automorphism groups of maps, surfaces and smarandache geometries, by linfan mao

8/7/2019 AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES, by LINFAN MAO

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AUTOMORPHISM GROUPS OF MAPS, SURFACES¸

AND SMARANDACHE GEOMETRIES¸

LINFAN MAO¸

Institute of Systems ScienceAcademy of Mathematics and SystemsChinese Academy of SciencesBeijing 100080, P.R.China

[email protected]

http://lanl.arxiv.org/abs/math/0505318v1









































Abstract:

A combinatorial map is a connected topological graph cellularly embedded in a sur-face. This monograph concentrates on the automorphism group of a map, which isrelated to the automorphism groups of a Klein surface and a Smarandache manifold,also applied to the enumeration of unrooted maps on orientable and non-orientablesurfaces. A number of results for the automorphism groups of maps, Klein surfacesand Smarandache manifolds and the enumeration of unrooted maps underlying a

graph on orientable and non-orientable surfaces are discovered. An elementary clas-sification for the closed s-manifolds is found. Open problems related the combi-natorial maps with the differential geometry, Riemann geometry and Smarandachegeometries are also presented in this monograph for the further applications of thecombinatorial maps to the classical mathematics.

AMS(2000) 05C15, 05C25, 20H15, 51D99, 51M05

i



Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Chapter 1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1 Klein surface and s- m a n i f o l d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Classification of Klein surfaces and s-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3§2 Map and embedding of a graph on surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The embedding of a graph on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Map and rooted map on surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Classification maps and embeddings of a graph on a surfaces. . . . . . . . . . . . . . . . . 82.5 Maps as a combinatorial model of Klein surfaces and s-manifolds .......... 10§3 The semi-arc automorphism group of a graph with application to maps enumer-a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 13.1 The semi-arc automorphism group of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 A scheme for enumerating maps underlying a graph. . . . . . . . . . . . . . . . . . . . . . . . .13§4 A relation among the total embeddings and rooted maps of a graph on genus 164.1 The rooted total map and embedding polynomial of a graph . . . . . . . . . . . . . . . . 16

4.2 The number of rooted maps underlying a graph on genus . . . . . . . . . . . . . . . . . . . 20Chapter 2 On the automorphisms of a Klein Surface and a s-manifold 26§1 An algebraic definition of a voltage map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261.1 Coverings of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2 Voltage maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28§2 Combinatorial conditions for a group being that of a map . . . . . . . . . . . . . . . . . . . 292.1 Combinatorial conditions for an automorphism group of a map ............. 302.2 The measures on a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34§3 A combinatorial refinement of Huriwtz theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37§4 The order of an automorphism of a Klein surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 The minimum genus of a fixed-free automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 The maximum order of an automorphism of a map .. . . . . . . . . . . . . . . . . . . . . . . . 45Chapter 3 On the Automorphisms of a graph on Surfaces . . . . . . . . . . . . . . 48§1 A necessary and sufficient condition for a group of a graph beingthat of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48§2 The automorphisms of a complete graph on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 53§3 The automorphisms of a semi-regular graph on surfaces . . . . . . . . . . . . . . . . . . . . . 61§4 The automorphisms of one face maps.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64Chapter 4 Application to the Enumeration of Unrooted Maps and s-Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68§1 The enumeration of unrooted complete maps on surfaces .................... 68



Contents iii

§2 The enumeration of a semi-regular graph on surfaces . . . . . . . . . . . . . . . . . . . . . . . . 74

§3 The enumeration of a bouquet on surfaces...................................77§4 A classification of the closed s- m a n i f o l d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2Chapter 5 Open Problems for Combinatorial Maps . . . . . . . . . . . . . . . . . . . . . . 8 85.1 The uniformization theorem for simple connected Riemann surfaces . . . . . . . . . 885.2 T he R i e mann- R oc h t he or e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Combinatorial construction of an algebraic curve of genus ..................895.4 Classification of s-manifolds by maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.5 Gauss mapping among surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.6 The Gauss-Bonnet theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.7 Riemann manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92



Chapter1

Preliminary

All surfaces are 2-dimensional compact manifolds without boundary, graphs areconnected, possibly with loops or multiple edges and groups are finite in the context.For terminology and notation not defined in this book can be seen in [33] , [34] and[35] for graphs and maps and in [6], [73] for groups.

§1. Klein surface and s-manifold

1.1 Definitions1.1.1 Definition of a Klein surface

The notion of Klein surface is introduced by Alling and Greenleaf [2] in 1971concerned with real algebraic curves, correspondence with that of Riemann surfaceconcerned with complex algebraic curves. For introducing this concept, it is needto enlarge analytic functions to those of dianalytic functions first.

Now let f : A −→ C be a mapping. Write z = x + iy,x,y ∈ R, i =√−1, z =

x − iy and f (z) = u(x, y) + iv(x, y) f (z) = u(x, y) − iv(x, y)for certain functionsu, v : A −→ R.

A mapping f : A−→ C

is analytic on A if ∂f

∂z

= 0 (Cauchy-Riemann equation )

and is antianlytic on A if ∂f ∂z = 0.

A mapping f is said to be dianalytic if its restriction to every connected compo-nent of A is analytic or antianalytic.

Now we can formally define a Klein surface.A Klein surface is a Hausdorff, connected, topological space S together with a

family

= (U i, φi) |i ∈ I such that the chart U i|i ∈ I is an open covering of S , each map φi : U i −→ Ai is a homeomorphism onto an open subset Ai of C orC+ = z ∈ C : Imz ≥ 0 and the transition functions

φij = φiφ−

j : φ j(U i U j ) −→ φi(U i

U j ).

are dianalytic.The family

is said to be a topological atlas on S .

The boundary of S is the set

∂S = x ∈ S |there exists i ∈ I, x ∈ U i, φi(x) ∈ Rand φi(U i) ⊆ C+.

The existence of Klein surfaces is obvious, for example, a Riemann surface is aKlein surface viewed as an orientable surface with empty boundary and

to be

analytic functions. Whence, we have the following relation:



Chapter 1 Preliminary 2

Riemann Sufaces ⊂ Klein surfaces.

The upper half plane H = z ∈ C|Imz > 0 with (U 1 = H, φ1 = 1H ) and theopen unit disc D = z ∈ C||z| < 1 with (U 1 = d, φ1 = 1D) in C are two Kleinsurfaces with empty boundary and analytic structures.

If k(S ), g(S ) and χ(S ) are the number of connected components of ∂S , thetopological genus and the Euler characteristic of a surface S , then we have that

Theorem 1.1.1([2])

χ(S ) = 2 − 2g(S ) − k(S ), if S is orientable,

2 − g(S ) − k(S ). if S is non − orientable.

1.1.2 Definition of a Smarandache geometry

By the history, we know that classical geometries include the Euclid geometry,the hyperbolical geometry and the Riemann’s geometry. Each of the later two isproposed by denial the 5th postulate for parallel lines in the Euclid postulates of geometry. The Smarandache geometries is proposed by Smarandache in 1969 ([61]),which is a generalization of the classical geometries, i.e., the Euclid, Lobachevshy-Bolyai-Gauss and Riemannian geometries may be united altogether in the samespace, by some Smarandache geometries. These last geometries can be either par-

tially Euclidean and partially Non-Euclidean, or Non-Euclidean. It seems that theSmarandache geometries are connected with the Relativity Theory (because they in-clude the Riemann geometry in a subspace) and with the Parallel Universes (becausethey combine separate spaces into one space) too([32]).

In [61], Smarandache defined several specific types of Smarandache geometries,such as the paradoxist geometry , the non-geometry , the counter-projective geometry and the anti-geometry . He also posed a question on the paradoxist geometry, i.e.,

find a nice model on manifolds for this paradoxist geometry and study some of itscharacteristics.

An axiom is said smarandachely denied if in the same space the axiom behavesdifferently (i.e., validated and invalided; or only invalided but in at least two district

ways).A Smarandache geometry is a geometry which has at least one smarandachely

denied axiom1∗ . At present, the Smarandache manifolds (abbreviated s-manifolds)are the central object discussed in the Smarandache geometries today. More resultsfor the Smarandache geometries can be seen in the references [4], [16],[27]− [28], [32]and [58] − [59] etc..

The idea of an s-manifold was based on a hyperbolic paper in [69] and creditedto W.Thurston. A more general idea can be found in [59]. According to the survey[27] of H.Iseri, an s-manifold is combinatorially defined as follows:

1Also see www.gallup.unm.edu/˜ samrandache/geometries.htm




An s-manifold is any collection C

(T, n) of these equilateral triangular disks T i, 1

≤i ≤ n satisfying the following conditions:(i) Each edge e is the identification of at most two edges ei, e j in two distinct

triangular disks T i, T j, 1 ≤ i, j ≤ n and i = j;(ii) Each vertex v is the identification of one vertex in each of five, six or seven

distinct triangular disks.The vertices are classified by the number of the disks around them. A vertex

around five, six or seven triangular disks is called an elliptic vertex , a Euclid vertex or a hyperbolic vertex , respectively.

An s-manifold is called closed if the number of triangular disks is finite and eachedge is shared by exactly two triangular disks, each vertex is completely around by

triangular disks. It is obvious that a closed s-manifold is a surface and its Eulercharacteristic can be defined by the Theorem 1.1.1.

1.2 Classification of Klein surfaces and s-manifolds

A morphism between the Klein surfaces S and S ′ is a continuous map f : S → S ′

such that f (∂S ) ⊆ ∂S ′ and given s ∈ S , there exist chart (U, φ) and (V, ψ) at s andf (s) respectively, and an analytic function F : φ(U ) → C such that

ψ(f (s)) = Φ(F (φ(s))),

where, Φ :

C → C+ : x + iy

→x + i

|y

|is a continuous map.

An automorphism of a Klein surface S is an 1 − 1 morphism f : S → S . It hasbeen known that for a given Klein surface S , the set AutS of automorphisms of S forms a group with respect to the composition operation and AutH = P GL(2, R).

Let Γ be a discrete subgroup of AutH . We say that Γ is a non-euclidean crys-tallographic group( shortly NEC group) if the quotient H/Γ is compact.

More results can be seen in [11]. Typical results for automorphisms of a Kleinsurface S are as follows.

Theorem 1.1.2([11]) Let S be a compact Klein surface, g = g(S ) and k = k(S ),then

( i) there exists an NEC group Γ such that AutS

∼= N Ω(Γ)/Γ, where Ω = AutH .

( ii) if S satisfies the condition 2g + k ≥ 3 if S is orientable and g + k ≥ 3 if S is non-orientable, then AutS is finite.

Similarly, two s-manifolds C1(T, n) and C2(T, n) are called to be isomorphic if there is an 1 − 1 mapping τ : C1(T, n) → C2(T, n) such that for ∀T 1, T 2 ∈ C1(T, n),

τ (T 1

T 2) = τ (T 1)

τ (T 2).

If C1(T, n) = C1(T, n) = C(T, n), τ is called an automorphism of the s-manifoldC(T, n). All automorphisms of an s-manifold form a group under the compositionoperation, called the automorphism group of an s-manifold C(T, n), denoted byAut

C(T, n).




§2. Map and embedding of a graph on surface

2.1 Graphs

Combinatorially, a graph Γ is a 2-tuple (V, E ) consists of a finite non-emptyset V of vertices together with a set E of unordered pairs of vertices, i.e., E ⊆V × V ([22], [35], [70]). Often denoted by V (Γ), E (Γ) the vertex set and edge set of the graph Γ.

The cardinal numbers of |V | and |E | are called the order and the size of thegraph Γ.

We can also obtain a representation of a graph Γ representing a vertex u by apoint p(u), p(u)

= p(v) if u

= v and an edge (u, v) by a curve connecting the points

p(u) and p(v) on the plane.For example, the graph in the Fig. 1.1

Fig 1.1¸

is a graph Γ = (V, E ) with V = u,v,w,x and

E = (u, u), (v, v), (w, w), (x, x), (u, v), (v, w), (w, x), (x, u).

A walk of a graph Γ is an alternating sequence of vertices and edges u1, e1, u2, e2,

· · · , en, un1 with ei = (ui, ui+1) for 1 ≤ i ≤ n. The number n is the length of thewalk. If u1 = un+1, the walk is said to be closed , and open otherwise. For example,ue1ve2we6we3xe3we2v is a walk in the graph of the Fig. 1.1. A walk is called a trail if all its edges are distinct and a path if all the vertices are distinct. A closed pathis said to be a circuit.

A graph Γ is connected if there is paths connecting any two vertices in this graphand is simple if any 2-tuple (u, v) ∈ E (Γ) ⊆ V (Γ) × V (Γ) appears once at most andu = v.

Let Γ be a graph. For ∀u ∈ V (Γ), the neighborhood N vΓ(u) is defined by N vΓ(u) =v|(u, v) or (v, u) ∈ E (Γ). Its cardinal |N vΓ(u)| is called the valency of the vertexu in the graph Γ, denoted by ρΓ(u). By the enumeration of edges, we know the




following result

u ∈ V (Γ)ρΓ(u) = 2|E (Γ)|.

2.2 The embedding of a graph on surfaces

A map on a surface S is a kind of partition S which enables us to obtain home-omorphisms of 2-cells (x, y)|x2 + y2 < 1, x , y ∈ R if we remove from S all thecurves used to partite S . There is a classical result for the partition of a surfacegotten by T.Rado in 1925.

Theorem 1.2.1([52]) For any compact surface S , there exist a triangulation T i, i ≥1 on S .

This theorem is fundamental for the topological graph theory, which enables usto discussion a surface combinatorially.

For any connected graph Γ = (V (Γ), E (Γ)) and a surface S , an embedding of thegraph Γ in the surface S is geometrical defined to be a continuous 1 − 1 mappingτ : Γ → S . The image τ (Γ) is contained in the 1-skeleton of a triangulation of thesurface S . If each component in S − τ (Γ) homeomorphic to an open disk, then theembedding is said a 2-cell embedding, where, components in S − τ (Γ) are calledfaces. All embeddings considered in this book are 2-cell embeddings.

Let Γ be a graph. For v ∈ V (Γ), denote by N eΓ(v) = e1, e2, · · · , eρ(v) all theedges incident with the vertex v. A permutation on e1, e2, · · · , eρ(v) is said a purerotation . All pure rotations incident a vertex v is denoted by (v). A pure rotation system of the graph Γ is defined to be

ρ(Γ) = (v)|v ∈ V (Γ)and all pure rotation systems of the graph Γ is denoted by (Γ).

The first characteristic for embedding of a graph on orientable surfaces is foundby Heffter in 1891 and formulated by Edmonds in 1962, states as follows.

Theorem 1.2.2([17]) Every pure rotation system for a graph Γ induces a uniqueembedding of Γ into an orientable surface. Conversely, every embedding of a graph Γ into an orientable surface induces a unique pure rotation system of Γ.

According to this theorem, we know that the number of orientable embeddingsof a graph Γ is

v∈V (Γ)(ρ(v) − 1)!.

The characteristic for embedding of a graph on locally orientable surface is usedby Ringel in the 1950s and gave a formal proof by Stahl in 1978([22][62]).

From the topological theory, embedded vertex and face can be viewed as disk,and an embedded edge can be viewed as an 1-band which is defined as a topologicalspace B together with a homeomorphism h : I × I → B, where I = [0, 1], the unitinterval.




Define a rotation system ρL(Γ) to be a pair (J

, λ), whereJ

is a pure rotationsystem of Γ, and λ : E (Γ) → Z 2. The edge with λ(e) = 0 or λ(e) = 1 is called type0 or type 1 edge, respectively. The rotation system of a graph Γ are defined by

L(Γ) = (J , λ)|J ∈ (Γ), λ : E (Γ) → Z 2.

Then we know that

Theorem 1.2.3([22][62]) Every rotation system on a graph Γ defines a unique locally orientable embedding of Γ → S . Conversely, every embedding of a graph Γ → S defines a rotation system for Γ.

For any embedding of the graph Γ, there is a spanning tree T such that everyedge on this tree is type 0([43]). Whence the number of embeddings of a graph Γon locally orientable surfaces is

2β (Γ)

v∈V (Γ)

(ρ(v) − 1)!

and the number of embeddings of Γ on the non-orientable surfaces is

(2β (Γ) − 1)

v∈V (Γ)

(ρ(v) − 1)!.

The following result is the Euler-Poincare formula for an embedding of a graphon surface.

Theorem 1.2.4 If a graph Γ can be embedded into a surface S , then

ν (Γ) − ε(Γ) + φ(Γ) = χ(S ),

where, ν (Γ), ε(Γ) and φ(Γ) are the order, size and the number of faces of the graph Γ, and χ(S ) is the Euler characteristic of the surface S :

χ(S ) =

2 − 2 p, if S is orientable,2

−q, if S is non

−orientable.

2.3. Map and rooted map on surface

In 1973, Tutte gave an algebraic representation for an embedding of a graph onlocally orientable surface ([66], which transfer a geometrical partition of a surfaceto a kind of permutation in algebra.

According to the summary in [33], a map M = (X α,β , P ) is defined to be a basicpermutation P , i.e, for any x ∈ X α,β , no integer k exists such that P kx = αx,acting on X α,β , the disjoint union of quadricells Kx of x ∈ X (the base set), whereK = 1,α,β ,αβ is the Klein group, satisfying the following two conditions:




( Ci) αP

=P −1α;

( Cii) the group ΨJ =< α, β, P > is transitive on X α,β .

For a given map M = (X α,β , P ), it can be shown that M ∗ = (X β,α, P αβ ) is alsoa map, call it the dual of the map M . The vertices of M are defined as the pairs of conjugatcy orbits of P action on X α,β by the condition (Ci) and edges the orbits of K on X α,β , for example,∀x ∈ X α,β , x,αx,βx,αβx is an edge of the map M . Definethe faces of M to be the vertices in the dual map M ∗. Then the Euler characteristicχ(M ) of the map M is

χ(M ) = ν (M ) − ε(M ) + φ(M )

where,ν (M ), ε(M ), φ(M ) are the number of vertices, edges and faces of the map M ,respectively.

For example, the graph K 4 on the tours with one face length 4 and another 8 ,shown in the Fig. 1.2, can be algebraic represented as follows:

A map (X α,β , P ) with X α,β = x,y,z,u,v,w,αx,αy,αz,αu,αv,αw,βx,βy,βz,βu,βv,βw,αβx,αβy,αβz,αβu,αβv,αβw and

P = (x,y,z)(αβx,u,w)(αβz,αβu,v)(αβy,αβv,αβw)

× (αx,αz,αy)(βx,αw,αu)(βz,αv,βu)(βy,βw,βv)

The four vertices of this map are (x,y,z), (αx,αz,αy), (αβx,u,w), (βx,αw,αu),(αβz,αβu,v), (βz,αv,βu) and (αβy,αβv,αβw), (βy,βw,βv) and six edges aree,αe,βe,αβe, where, e ∈ x,y,z,u,v,w. The Euler characteristic χ(M ) isχ(M ) = 4 − 6 + 2 = 0.

Fig 1.2¸

Geometrically, an embedding M of a graph Γ on a surface is a map and has analgebraic representation. The graph Γ is said the underlying graph of the map M




and denoted by Γ = Γ(M ). For determining a given map (X α,β

,P

) is orientable ornot, the following condition is needed.

( Ciii) If the group ΨI =< αβ, P > is transitive on X α,β , then M is non-orientable. Otherwise, orientable.

It can be shown that the number of orbits of the group Ψ I =< αβ, P > inthe Fig.1.1 action on X α,β = x, y,z, u, v,w, αx, αy, αz,αu, αv,αw, βx, βy, βz,βu,βv,βw,αβx,αβy,αβz,αβu,αβv,αβw is 2. Whence, it is an orientable map andthe genus of the surface is 1. Therefore, the algebraic representation is correspondentwith its geometrical mean.

A rooted map M x is a map M such that one quadricell x ∈ X α,β is marked

beforehand, which is introduced by Tutte in the enumeration of planar maps. Theimportance of the root is to destroy the symmetry in a map. That is the reason whywe can enumerate rooted maps on surfaces by combinatorial approaches.

2.4. Classification maps and embeddings of a graph on surfaces

2.4.1 Equivalent embeddings of a graph

From references, such as, [22], [70], etc., two embeddings (J 1, λ1), (J 2, λ2) of Γ onan orientable surface S are called equivalent if there exists an orientation-preservinghomeomorphism τ of the surface S such that τ : J 1 → J 2, and τ λ = λτ . If (

J 1, λ1) = (

J 2, λ2) = (

J , λ), then an orientation-preserving homeomorphism map-

ping (J 1, λ1) to (J 2, λ2) is called an automorphism of the embedding (J , λ). Cer-tainly, all automorphisms of an embedding form a group, denoted by Aut(J , λ).

Enumerating the non-equivalent orientable embeddings of a complete graph anda complete bipartite graph are considered by Biggs, White, Mull and Rieper etal in [6], [54] − [55]. Their approach is generalized in the following Section 2.3.2for enumerating non-equivalent embeddings of a given graph on locally orientablesurface in the view of maps on surfaces.

2.4.2 Isomorphism of maps

Two maps M 1 = (X 1α,β , P 1) and M 2 = (X 2α,β , P 2) are said to be isomorphic if

there exists a bijection ξ

ξ : X 1α,β −→ X 2α,β

such that for ∀x ∈ X 1α,β ,

ξα(x) = αξ(x), ξβ (x) = βξ (x) and ξP 1(x) = P 2ξ(x).

Call ξ an isomorphism between M 1 and M 2. If M 1 = M 2 = M , then an isomorphismbetween M 1 and M 2 is called an automorphism of M . All automorphisms of a mapM form a group, called the automorphism group of M and denoted by AutM .




Similarly, two rooted maps M x1

, M y

2are said to be isomorphic if there is an

isomorphism θ between them such that θ(x) = y. All automorphisms of a rootedmap M r also form a group, denoted by AutM r. It has been known that AutM r istrivial ([33]).

Using isomorphisms between maps, an alternative approach for determiningequivalent embeddings of a graph on locally orientable surfaces can be gotten, whichhas been used in [43], [49]−[50] for determining the number of non-equivalent embed-dings of a complete graph, a semi-regular graph and a Cayley graph Γ = Cay(G : S )with AutΓ ∼= R(G) × H , is defined as follows.

For a given map M underlying a graph Γ, it is obvious that AutM |Γ ≤ AutΓ.We extend the action ∀g ∈ AutΓ on V (Γ) to X α,β , where X = E (Γ), as follows:

∀x ∈ X α,β , if xg = y, then define (αx)g = αy, (βx)g = βy and (αβx)g = αβy.Two maps (embeddings) M 1, M 2 with a given underlying graph Γ are equivalent

if there exists an isomorphism ζ between them induced by an element ξ. Call ζ an equivalence between M 1, M 2. Certainly, on an orientable surface, an equivalencepreserve the orientation on this surface.

Theorem 1.2.5 Let M = (X α,β , P ) be a map with an underlying graph Γ, ∀g ∈ AutΓ.Then the extend action of g on X α,β with X = E (Γ) is an automorphism of map M iff ∀v ∈ V (M ), g preserves the cyclic order of v.

Proof Assume that ζ ∈ AutM is induced by the extend action of an automor-phism g in Γ, u, v

∈V (M ) and ug = v. Not loss of the generality, we assume

that

u = (x1, x2, · · · , xρ(u))(αxρ(u), · · · , αx2, αx1)

v = (y1, y2, · · · , yρ(v))(αyρ(v), · · · , αy2, αy1).

Without loss of generality , we can assume that

(x1, x2, · · · , xρ(u))g = (y1, y2, · · · , yρ(v)),

that is,

(g(x1), g(x2), · · · , g(xρ(u))) = (y1, y2, · · · , yρ(v)).

Whence, g preserves the cyclic order of vertices in the map M .On the other hand, if the extend action of g ∈ AutΓ on X α,β preserves the cyclic

order of each vertex in M , i.e., ∀u ∈ V (Γ), ∃v ∈ V (Γ) such that ug = v. Assumethat

P =

u∈V (M )

u.

Then




P g = u∈V (M )

ug = v∈V (M )

v = P .

Therefore, the extend action of g on X α,β is an automorphism of the map M .

2.5 Maps as a combinatorial model of Klein surfaces and s-manifolds

2.5.1 The model of Klein surfaces

Given a complex algebraic curve, it is a very important problem to determine itsbirational automorphisms. For curve C of genus g ≥ 2, Schwarz proved that Aut(C)is finite in 1879 and Hurwitz proved

|Aut(

C)

| ≤84(g

−1)(see [18] ). As observed by

Riemann, groups of birational automorphisms of complex algebraic curves are thesame as the automorphism groups of the compact Riemann surfaces. The latter canbe combinatorially dealt with the approach of maps.

Theorem 1.2.6([8][29]) If M is an orientable map of genus p, then AutM is iso-morphic to a group of conformal transformations of a Riemann surface of genus

p.

According to the Theorem 1.1.2, the automorphism group of a Klein surface hasthe same form as a Riemann surface. Similar to the proof of the Theorem 5.6 in[29], we can get a result similar to the Theorem 1.2.6 for Klein surfaces.

Theorem 1.2.7 If M is a locally orientable map of genus q, then AutM is isomorphicto a group of conformal transformations of a Klein surface of genus q.

Proof By a result in [8], AutM ∼= N T (A)/A, Where T =< a,b,c|a2 = b2 = c2 =(ba)2 = (ac)m = (cb)n = 1 >, A ≤ T and T can be realized by an automorphismgroup of a tessellation on the upper plane, A a NEC subgroup. According to theTheorem 1.1.2, The underlying surface S of M has S = H/A with Ω = AutH =P GL(2, R) being the automorphism group of the upper half plane H . Since T ≤ Ω,we know that AutM ∼= N T (A)/A ≤ N Ω(A)/A, isomorphic to a group of conformaltransformations of the Klein surface S = H/G.

2.5.2 The model of closed s-manifolds

For a closed s-manifold C(T, n), we can define a map M by V (M ) = the vertices inC(T, n), E (M ) = the edges in C(T, n) and F (M ) = T, T ∈ C(T, n). Then, M is a triangular map with vertex valency ∈ 5, 6, 7. On the other hand, if M is atriangular map on surface with vertex valency ∈ 5, 6, 7, we can define C(T, φ(M ))by

C(T, φ(M )) = f |f ∈ F (M ).

Then, C(T, φ(M )) is an s-manifold. Therefore, we get the following result.




Theorem 1.2.8 Let C(T, n),

M(T, n) and

M∗(T, n) be the set of s-manifolds with

n triangular disks, triangular maps with n faces and vertex valency ∈ 5, 6, 7 and cubic maps of order n with face valency ∈ 5, 6, 7. Then

(i) There is a bijection between M(T, n) and C(T, n);

(ii) There is also a bijection between M∗(T, n) and C(T, n).

§3. The semi-arc automorphism group of a graph with application tomaps enumeration

3.1 The semi-arc automorphism group of a graph

Let Γ be a graph with vertex set V (Γ) and edge set E (Γ). By the definition, anautomorphism of Γ on V (Γ)

E (Γ) is an 1 − 1 mapping (ξ, η) on Γ such that

ξ : V (Γ) → V (Γ), η : E (Γ) → E (Γ),

satisfying that for any incident elements e, f , (ξ, η)(e) and (ξ, η)(f ) are also incident.Certainly, all automorphisms of a graph Γ form a group, denoted by AutΓ.

Now an edge e = uv ∈ E (Γ) can be divided into two semi-arcs eu, ev. Call uthe root vertex in the semi-arc eu. Two semi-arc eu, f v are said incident if u = vor e = f . The set of all semi-arcs of a graph Γ is denoted by X 1

2(Γ). A semi-arc

automorphism of a graph, first appeared in [43] and then applied to the enumeration

rooted maps on surfaces underlying a graph Γ in [46], is defined as follows.Definition 1.3.1 An 1 − 1 mapping ξ on X 1

2(Γ) such that ∀eu, f v ∈ X 1

2(Γ), ξ(eu)

and ξ(f v) are incident if eu and f v are incident, is called a semi-arc automorphism of the graph Γ.

All semi-arc automorphisms of a graph also form a group under the compositionoperation, denoted by Aut 1

2Γ, which is more important for the enumeration of maps

on surfaces and also important for determine the conformal transformations on aKlein surface. The following table lists semi-arc automorphism groups of some well-known graphs, which give us some useful information for the semi-arc automorphismgroups, for example, Aut 1

2K n = S n but Aut 1

2Bn = S n[S 2]

= AutBn.

Γ Aut12

Γ order

K n S n n!K m,n(m = n) S m × S n m!n!

K n,n S 2[S n] 2n!2

Bn S n[S 2] 2nn!Dpn S 2 × S n 2n!

Dpk,ln (k = l) S 2[S k] × S n × S 2[S l] 2k+ln!k!l!Dpk,k

n S 2 × S n × (S 2[S k])2 22k+1n!k!2

table 3.1¸




Here, Dpn

is a dipole graph with 2 vertices, n multiple edges and Dpk,l

nis a general-

ized dipole graph with 2 vertices, n multiple edges, and one vertex with k bouquetsand another, l bouquets. Comparing the semi-arc automorphism groups in the sec-ond column with automorphism groups of graphs in the first column in table 3.1,it is easy to note that the semi-arc automorphism group are the same as the auto-morphism group in the first two cases. In fact, it is so by the following Theorem1.3.1.

For ∀g ∈ AutΓ, there is also an induced action g| 12 on X 12

(Γ), g : X 12

(Γ) →X 1

2(Γ), as follows:

∀eu

∈X 1

2(Γ), g(eu) = (g(e)g(u).

All induced action of the elements in AutΓ on X 12

(Γ) is denoted by AutΓ| 12 . Noticethat

AutΓ ∼= AutΓ| 12 .

We have the following result.

Theorem 1.3.1 For a graph Γ without loops,

Aut 12

Γ = AutΓ| 12 .

Proof By the definition, we only need to prove that for ∀ξ 12

∈ Aut 12

Γ, ξ = ξ12|Γ ∈

AutΓ and ξ12

= ξ| 12 . In fact, for any ∀eu, f x ∈ X 12

(Γ), where, e = uv ∈ E (Γ) and

f = xy ∈ E (Γ), if

ξ 12

(eu) = f x,

then by the definition, we know that

ξ 12

(ev) = f y.

Whence, ξ1

2

(e) = f . That is, ξ 1

2 |Γ

∈AutΓ.

Now since there is not a loop in Γ, we know that ξ 12|Γ = idΓ if and only if

ξ12

= idΓ. Therefore, ξ 12

is induced by ξ 12|Γ on X 1

2(Γ), that is,

Aut12

Γ = AutΓ| 12 .

Notice that for a given graph Γ, X 12

(Γ) = X β , if we equal eu to e and ev to βe

for an edge e = uv ∈ E (Γ).For a given map M = (X α,β , P ) underlying a graph Γ, we have known that

AutM |Γ ≤ AutΓ, which made us to extend the action of an automorphism g of the graph Γ on X α,β with X = E (Γ) to get automorphisms of a map induced byautomorphisms of its underlying graph. More detail, we can get the following result.




Theorem 1.3.2 Two maps M 1

= (X α,β

,P 1

) and M 2

= (X α,β

,P 2

) underlying a graph Γ are

( i) equivalent iff there is an element ζ ∈ Aut12

Γ such that P ζ 1 = P 2 and

( ii)isomorphic iff there is an element ζ ∈ Aut 12

Γ such that P ζ 1 = P 2 or P ζ

1 =

P −12 .

Proof By the definition of equivalence between maps, if κ is an equivalence be-tween M 1 and M 2, then κ is an isomorphism between M 1 and M 2 induced by anautomorphism ι ∈ AutΓ. Notice that

AutΓ ∼= AutΓ

|

12

≤Aut 1

2Γ.

Whence, we know that ι ∈ Aut 12

Γ.

Now if there is a ζ ∈ Aut 12

Γ such that P ζ 1 = P 2, then ∀ex ∈ X 1

2(Γ), ζ (ex) =

ζ (e)ζ (x). Now assume that e = (x, y) ∈ E (Γ), then by our convention, we know thatif ex = e ∈ X α,β , then ey = βe. Now by the definition of an automorphism on thesemi-arc set X 1

2(Γ), if ζ (ex) = f u, where f = (u, v), then there must be ζ (ey) = f v.

Notice that X 12

(Γ) = X β . We know that ζ (ey) = ζ (βe) = βf = f v. We can also

extend the action of ζ on X 12

(Γ) to X α,β by ζ (αe) = αζ (e). Whence, we know that∀e ∈ X α,β ,

αζ (e) = ζα(e), βζ (e) = ζβ (e) and

P ζ 1 (e) =

P 2(e).

Therefore, the extend action of ζ on X α,β is an isomorphism between the map M 1and M 2. Whence, ζ is an equivalence between the map M 1 and M 2. So the assertionin (i) is true.

For the assertion in (ii), if there is an element ζ ∈ Aut 12

Γ such that P ζ 1 = P 2,

then the map M 1 is isomorphic to M 2. If P ζ 1 = P −12 , then P ζα

1 = P 2. The map M 1is also isomorphic to M 2. This is the sufficiency of (ii).

By the definition of an isomorphism ξ between maps M 1 and M 2, we know that∀x ∈ X α,β ,

αξ(x) = ξα(x), βξ (x) = ξβ (x) andP

ξ1(x) =

P 2(x).

By the convention, the condition

βξ (x) = ξβ (x) and P ξ1(x) = P 2(x).

is just the condition of an automorphism ξ or αξ on X 12

(Γ). Whence, the assertion

in (ii) is also true.

3.2 A scheme for enumerating maps underlying a graph

For a given graph Γ, denoted by E O(Γ), E N (Γ) and E L(Γ) the sets of embeddingsof Γ on the orientable surfaces, on the non-orientable surfaces and on the locally




orientable surfaces, respectively. For determining the number of non-equivalentembeddings of a graph on surfaces and non-isomorphic unrooted maps underlying agraph, another form of the Theorem 1.3.2 using the group action idea is need, whichis stated as follows.

Theorem 1.3.3 For two maps M 1 = (X α,β , P 1) and M 2 = (X α,β , P 2) underlying a graph Γ, then

( i) M 1, M 2 are equivalent iff M 1, M 2 are in one orbits of Aut 12

Γ action on X 12

(Γ);

( ii) M 1, M 2 are isomorphic iff M 1, M 2 are in one orbits of Aut 12

Γ× < α > action on X α,β .

Now we can established a scheme for enumerating the number of non-isomorphicunrooted maps and non-equivalent embeddings in a given set of embeddings of agraph on surfaces by using the Burnside Lemma as the following.

Theorem 1.3.4 For a given graph Γ, let E ⊂ E L(Γ), then the numbers n(E , Γ) and η(E , Γ) of non-isomorphic unrooted maps and non-equivalent embeddings in E arerespective

n(E , Γ) =1

2|Aut 12

Γ|

g∈Aut12Γ

|Φ1(g)|,

where, Φ1(g) =

P|P ∈ E and

P g =

P or

P gα =

Pand

η(E , Γ) =1

|Aut 12

Γ|

g∈Aut12Γ

|Φ2(g)|,

where, Φ2(g) = P|P ∈ E and P g = P.

Proof Define the group H = Aut 12

Γ× < α >. Then by the Burnside Lemmaand the Theorem 1.3.3, we get that

n(E , Γ) =1

|H|g∈H

|Φ1(g)|,

where, Φ1(g) = P|P ∈ E and P g = P. Now |H| = 2|Aut 12

Γ|. Notice that if P g = P , then P gα = P , and if P gα = P , then P g = P . Whence, Φ1(g)

Φ1(gα) = ∅.

We have that

n(E , Γ) =1

2|Aut 12

Γ|

g∈Aut12Γ

|Φ1(g)|,

where, Φ1(g) = P|P ∈ E and P g = P or P gα = P.A similar proof enables us to obtain that




η(E , Γ) = 1|Aut 1

2Γ|

g∈Aut1

2Γ

|Φ2(g)|,

where, Φ2(g) = P|P ∈ E and P g = P. From the Theorem 1.3.4, we get the following results.

Corollary 1.3.1 The numbers nO(Γ), nN (Γ) and nL(Γ) of non-isomorphic unrooted orientable maps ,non-orientable maps and locally orientable maps underlying a graph Γ are respective

nO(Γ) =1

2|Aut 12 Γ| g∈Aut1

2Γ |

ΦO

1(g)

|; (1.3.1)

nN (Γ) =1

2|Aut 12

Γ|

g∈Aut12Γ

|ΦN 1 (g)|; (1.3.2)

nL(Γ) =1

2|Aut 12

Γ|

g∈Aut12Γ

|ΦL1 (g)|, (1.3.3)

where, ΦO1 (g) = P|P ∈ E O(Γ) and P g = P or P gα = P, ΦN

1 (g) = P|P ∈ E N (Γ)and P g = P or P gα = P, ΦL

1 (g) = P|P ∈ E L(Γ) and P g = P or P gα = P.

Corollary 1.3.2 The numbers ηO(Γ), ηN (Γ) and ηL(Γ) of non-equivalent embeddingsof a graph Γ on orientable ,non-orientable and locally orientable surfaces are respec-tive

ηO(Γ) =1

|Aut 12

Γ|

g∈Aut12Γ

|ΦO2 (g)|; (1.3.4)

ηN (Γ) =1

|Aut 12

Γ|

g∈Aut12Γ

|ΦN 2 (g)|; (1.3.5)

ηL

(Γ) =

1

|Aut 12

Γ| g∈Aut1

2Γ|Φ

L2 (g)|, (1.3.6)

where, ΦO2 (g) = P|P ∈ E O(Γ) and P g = P, ΦN

2 (g) = P|P ∈ E N (Γ) and P g =P, ΦL

2 (g) = P|P ∈ E L(Γ) and P g = P.

For a simple graph Γ, since Aut 12

Γ = AutΓ by the Theorem 1.3.1, the formula

(1.3.4) is just the scheme used for counting the non-equivalent embeddings of acomplete graph, a complete bipartite graph in the references [6], [54] − [55], [70]. Foran asymmetric graph Γ, that is, Aut 1

2Γ = idΓ, we get the numbers of non-isomorphic

maps underlying a graph Γ and the numbers of non-equivalent embeddings of thegraph Γ by the Corollary 1.3.1 and 1.3.2 as follows.




Theorem 1.3.5 The numbers nO(Γ), nN (Γ) and nL(Γ) of non-isomorphic unrooted maps on orientable, non-orientable surface or locally surface with an asymmetricunderlying graph Γ are respective

nO(Γ) =

v∈V (Γ)

(ρ(v) − 1)!

2,

nL(Γ) = 2β (Γ)−1

v∈V (Γ)

(ρ(v) − 1)!

and

nN (Γ) = (2β (Γ)−1 − 12

) v∈V (Γ)

(ρ(v) − 1)!,

where, β (Γ) is the Betti number of the graph Γ.The numbers ηO(Γ), ηN (Γ) and ηL(Γ) of non-equivalent embeddings of an asym-

metric underlying graph Γ are respective

ηO(Γ) =

v∈V (Γ)

(ρ(v) − 1)!,

ηL(Γ) = 2β (Γ)

v∈V (Γ)

(ρ(v) − 1)!

and

ηN (Γ) = (2β (Γ) − 1)

v∈V (Γ)

(ρ(v) − 1)!.

§4. A relation among the total embeddings and rooted maps of agraph on genus

4.1 The rooted total map and embedding polynomial of a graph

For a given graph Γ with maximum valency ≥ 3, assume that ri(Γ), ri(Γ), i ≥ 0are respectively the numbers of rooted maps with an underlying graph Γ on theorientable surface with genus γ (Γ) + i − 1 and on the non-orientable surface withgenus γ (Γ) + i − 1, where γ (Γ) and γ (Γ) denote the minimum orientable genusand minimum non-orientable genus of the graph Γ, respectively. Define its rooted orientable map polynomial r[Γ](x) , rooted non-orientable map polynomial r[Γ](x)and rooted total map polynomial R[Γ](x) on genus by:




r[Γ](x) = i≥0

ri(Γ)xi,

r[Γ](x) =i≥0

ri(Γ)xi

and

R[Γ](x) =i≥0

ri(Γ)xi +i≥1

ri(Γ)x−i.

The total number of orientable embeddings of Γ is

d∈D(Γ)(d − 1)! and non-

orientable embeddings is (2β (Γ)−1) d∈D(Γ)

(d−1)!, where D(Γ) is its valency sequence.

Now let gi(Γ) and gi(Γ), i ≥ 0 respectively be the number of embeddings of Γ onthe orientable surface with genus γ (Γ)+ i−1 and on the non-orientable surface withgenus γ (Γ) + i − 1. The orientable genus polynomial g[Γ](x) , non-orientable genuspolynomial g[Γ](x) and total genus polynomial G[Γ](x) of Γ are defined by

g[Γ](x) =i≥0

gi(Γ)xi,

g[Γ](x) = i≥0 gi(Γ)xi

and

G[Γ](x) =i≥0

gi(Γ)xi +i≥1

gi(Γ)x−i.

The orientable genus polynomial g[Γ](x) is introduced by Gross and Furst in [23],and in [19], [23] − [24], the orientable genus polynomials of a necklace, a bouquet, aclosed-end ladder and a cobblestone are determined. The total genus polynomial isintroduced by Chern et al. in [13], and in [31], recurrence relations for the total genuspolynomial of a bouquet and a dipole are found. The rooted orientable map polyno-

mial is introduced in [43] − [44], [47] and the rooted non-orientable map polynomialin [48]. All the polynomials r[Γ](x), r[Γ](x), R[Γ](x) and g[Γ](x), g[Γ](x), G[Γ](x) arefinite by the properties of embeddings of Γ on surfaces.

Now we establish relations of r[Γ](x) with g[Γ](x), r[Γ](x) with g[Γ](x) andR[Γ](x) with G[Γ](x) as follows.

Lemma 1.4.1([25][45]) For a given map M , the number of non-isomorphic rooted maps by rooting on M is

4ε(M )

|AutM | ,




where ε(M ) is the number of edges in M .

Theorem 1.4.1 For a given graph Γ,

|Aut 12

Γ|r[Γ](x) = 2ε(Γ)g[Γ](x),

|Aut 12

Γ|r[Γ](x) = 2ε(Γ)g[Γ](x)

and

|Aut 12

Γ|R[Γ](x) = 2ε(Γ)G[Γ](x),

where Aut12 Γ and ε(Γ) denote the semi-arc automorphism group and the size of Γ,

respectively.

Proof For an integer k, denotes by Mk(Γ, S ) all the non-isomorphic unrootedmaps on an orientable surface S with genus γ (Γ)+ k − 1. According to the Lemma1.4.1, we know that

rk(Γ) =

M ∈Mk(Γ,S )

4ε(M )

|AutM |

=4ε(Γ)

|Aut 12

Γ× < α > | M ∈Mk(Γ,S )

|Aut 1

2

Γ

×< α >

||AutM | .

Since |Aut 12

Γ× < α > | = |(Aut 12

Γ× < α >)M ||M Aut1

2Γ×<α>| and |(Aut1

2Γ ×

< α >)M | = |AutM |, we have that

rk(Γ) =4ε(Γ)

|Aut 12

Γ× < α > |

M ∈Mk(Γ,S )

|M Aut1

2Γ×<α>|

=2ε(Γ)gk(Γ)

|Aut12 Γ|

.

Therefore, we get that

|Aut 12

Γ|r[Γ](x) = |Aut 12

Γ|i≥0

ri(Γ)xi

=i≥0

|Aut 12

Γ|ri(Γ)xi

=i≥0

2ε(Γ)gi(Γ)xi = 2ε(Γ)g[Γ](x).




Similarly, let Mk(Γ, S ) be all the non-isomorphic unrooted maps on an non-

orientable surface S with genus γ (Γ) + k − 1. Similar to the proof for orientablecase, we can get that

rk(Γ) =4ε(Γ)

|Aut 12

Γ× < α > |

M ∈Mk(Γ,S )

|Aut 12

Γ× < α > ||AutM |

=4ε(Γ)

|Aut 12

Γ× < α > |

M ∈

Mk(Γ,

S )

|M Aut1

2Γ×<α>|

=

2ε(Γ)gk(Γ)

|Aut12

Γ| .

Therefore, we also get that

|Aut 12

Γ|r[Γ](x) =i≥0

|Aut 12

Γ|ri(Γ)xi

=i≥0

2ε(Γ) gi(Γ)xi = 2ε(Γ)g[Γ](x).

Notice that

R[Γ](x) =i≥0

ri(Γ)xi +i≥1

ri(Γ)x−i

and

G[Γ](x) =i≥0

gi(Γ)xi +i≥1

gi(Γ)x−i.

By the previous discussion, we know that for k ≥ 0,

rk(Γ) =2ε(Γ)gk(Γ)

|Aut 1

2Γ

|and

rk(Γ) =

2ε(Γ)

gk(Γ)

|Aut 1

2Γ

|.


|Aut 12

Γ|R[Γ](x) = |Aut 12

Γ|(i≥0

ri(Γ)xi +i≥1

ri(Γ)x−i)

=i≥0

|Aut 12

Γ|ri(Γ)xi +i≥1

|Aut 12

Γ|ri(Γ)x−i

=i≥0

2ε(Γ)gi(Γ)xi +i≥0

2ε(Γ)gi(Γ)x−i = 2ε(Γ)G[Γ](x).

This completes the proof.




Corollary 1.4.1 Let be Γ a graph and s≥

0 be an integer. If rs(Γ) and g

s(Γ) are

the numbers of rooted maps underlying the graph Γ and embeddings of Γ on a locally orientable surface of genus s, respectively, then

|Aut12

Γ|rs( Γ) = 2ε(Γ)gs(Γ).

4.2 The number of rooted maps underlying a graph on genus

The Corollary 1.4.1 in the previous section can be used to find the implicitrelations among r[Γ](x),

r[Γ](x) or R[Γ](x) if the implicit relations among g[Γ](x),

g[Γ](x) or G[Γ](x) are known, and vice via.Denote the variable vector (x1, x2, · · ·) by x

¯,

r¯

(Γ) = (· · · , r2(Γ), r1(Γ), r0(Γ), r1(Γ), r2(Γ), · · ·),

g¯

(Γ) = (· · · , g2(Γ), g1(Γ), g0(Γ), g1(Γ), g2(Γ), · · ·).

The r¯

(Γ) and g¯

(Γ) are called the rooted map sequence and the embedding sequenceof the graph Γ.

Define a function F (x¯

, y¯

) to be y-linear if it can be represented as the followingform

F (x¯

, y¯

) = f (x1, x2, · · ·) + h(x1, x2, · · ·)i∈I

yi + l(x1, x2, · · ·) Λ∈O

Λ(y¯

),

where, I denotes a subset of index and O a set of linear operators. Notice thatf (x1, x2, · · ·) = F (x

¯, 0

¯), where 0

¯= (0, 0, · · ·). We have the following general result.

Theorem 1.4.2 Let G be a graph family and H ⊆ G. If their embedding sequencesg ¯

(Γ), Γ ∈ H, satisfies the equation

F H(x

¯

, g

¯

(Γ)) = 0, (4.1)

then its rooted map sequences r ¯

(Γ), Γ ∈ H satisfies the equation

F H(x ¯

,|Aut1

2Γ|

2ε(Γ)r

¯ (Γ)) = 0,

and vice via, if the rooted map sequences r ¯


F H(x ¯

, r ¯

(Γ)) = 0, (4.2)

then its embedding sequences g ¯





F H(x ¯

, 2ε(Γ)|Aut 1

2Γ|g

¯ (Γ)) = 0.

Even more, assume the function F (x ¯

, y ¯

) is y-linear and 2ε(Γ)|Aut 1

2Γ|

, Γ ∈ H is a constant.

If the embedding sequences g ¯

(Γ), Γ ∈ H satisfies the equation (4.1), then

F ⋄H(x ¯

, r ¯

(Γ)) = 0,

where F ⋄H(x ¯

, y ¯

) = F (x ¯

, y ¯

)+ ( 2ε(Γ)|Aut1

2Γ|

−1)F (x ¯

, 0 ¯

) and vice via, if its rooted map sequence

g ¯

(Γ), Γ ∈ H satisfies the equation (4.2), then

F ⋆H(x ¯

, g ¯

(Γ)) = 0.

where F ⋆H = F (x ¯

, y ¯

) + (|Aut1

2Γ|

2ε(Γ)− 1)F (x

¯ , 0 ¯

).

Proof According to the Corollary 1.4.1 in this chapter, for any integer s ≥ o andΓ ∈ H, we know that

|Aut12

Γ|rs( Γ) = 2ε(Γ)gs(Γ).

Whence,

rs(Γ) = 2ε(Γ)|Aut12

Γ| gs(Γ)

and

gs(Γ) =|Aut 1

2Γ|

2ε(Γ)rs(Γ).

Therefore, if the embedding sequences g¯

(Γ), Γ ∈ H satisfies the equation (4.1),then

F H(x,

|Aut 12

Γ|2ε(Γ) r(Γ)) = 0,

and vice via, if the rooted map sequences r¯

(Γ), Γ ∈ H satisfies the equation (4.2),then

F H(x¯

,2ε(Γ)

|Aut 12

Γ|g¯

(Γ)) = 0.

Now assume that F H(x¯

, y¯

) is a y-linear function and has the following form

F H(x

¯

, y

¯

) = f (x1, x2,

· · ·) + h(x1, x2,

· · ·)i∈I

yi + l(x1, x2,

· · ·) Λ∈O Λ(y

¯

),




whereO

is a set of linear operators. If F H

(x¯

, g¯

(Γ)) = 0, that is

f (x1, x2, · · ·) + h(x1, x2, · · ·) i∈I,Γ∈H

gi(Γ) + l(x1, x2, · · ·) Λ∈O,Γ∈H

Λ(g¯

(Γ)) = 0,

we get that

f (x1, x2, · · ·) + h(x1, x2, · · ·) i∈I,Γ∈H

|Aut 12

Γ|2ε(Γ)

ri(Γ)

+ l(x1, x2, · · ·) Λ∈O,Γ∈HΛ( |Aut 1

2Γ

|2ε(Γ) r(Γ)) = 0.

Since Λ ∈ O is a linear operator and 2ε(Γ)|Aut1

2Γ|

, Γ ∈ H is a constant, we also have

f (x1, x2, · · ·) +|Aut1

2Γ|

2ε(Γ)h(x1, x2, · · ·)

i∈I,Γ∈H

ri(Γ)

+|Aut1

2Γ|

2ε(Γ)l(x1, x2, · · ·)

Λ∈O,Γ∈H

Λ(r¯

(Γ)) = 0,

that is,

2ε(Γ)

|Aut 12

Γ|f (x1, x2, · · ·) + h(x1, x2, · · ·) i∈I,Γ∈H

ri(Γ) + l(x1, x2, · · ·) Λ∈O,Γ∈H

Λ(r¯

(Γ)) = 0.


F ⋄H(x¯

, r¯

(Γ)) = 0.

Similarly, if

F H(x, r(Γ)) = 0,then we can also get that

F ⋆H(x¯

, g¯

(Γ)) = 0.


Corollary 1.4.2 Let G be a graph family and H ⊆ G. If the embedding sequenceg ¯

(Γ) of a graph Γ ∈ G satisfies a recursive relation

i∈J,Γ∈H

a(i, Γ)gi(Γ) = 0,




where J is the set of index, then the rooted map sequence r ¯

(Γ) satisfies a recursiverelation

i∈J,Γ∈H

a(i, Γ)|Aut 12

Γ|2ε(Γ)

ri(Γ) = 0,

and vice via.

A typical example of the Corollary 1.4.2 is the graph family bouquets Bn, n ≥1. Notice that in [24], the following recursive relation for the number gm(n) of embeddings of a bouquet Bn on an orientable surface with genus m for n ≥ 2 wasfound.

(n + 1)gm(n) = 4(2n − 1)(2n − 3)(n − 1)2(n − 2)gm−1(n − 2)

+ 4(2n − 1)(n − 1)gm(n − 1).

and with the boundary conditions

gm(n) = 0 if m ≤ 0 or n ≤ 0;g0(0) = g0(1) = 1 and gm(0) = gm(1) = 0 for m ≥ 0;g0(2) = 4, g1(2) = 2, gm(2) = 0 for m ≥ 1.Since |Aut 1

2Bn| = 2nn!, we get the following recursive relation for the number

rm(n) of rooted maps on an orientable surface of genus m underlying a graph Bn bythe Corollary 1.4.2

(n2 − 1)(n − 2)rm(n) = (2n − 1)(2n − 3)(n − 1)2(n − 2)rm−1(n − 2)

+ 2(2n − 1)(n − 1)(n − 2)rm(n − 1),

and with the boundary conditionsrm(n) = 0 if m ≤ 0 or n ≤ 0;r0(0) = r0(1) = 1 and rm(0) = rm(1) = 0 for m ≥ 0;r0(2) = 2, r1(2) = 1, gm(2) = 0 for m

≥1.

Corollary 1.4.3 Let G be a graph family and H ⊆ G . If the embedding sequencesg ¯

(Γ), Γ ∈ G satisfies an operator equation

Λ∈O,Γ∈H

Λ(g ¯

(Γ)) = 0,

where O denotes a set of linear operators, then the rooted map sequences r ¯

(Γ), Γ ∈ Hsatisfies an operator equation

Λ∈O,Γ∈H

Λ(|Aut1

2Γ|

2ε(Γ)r

¯ (Γ)) = 0,




and vice via.

Let θ = (θ1, θ2, · · · , θk) ⊢ 2n, i.e.,k

j=1θ j = 2n with positive integers θ j. Kwak and

Shim introduced three linear operators Γ, Θ and ∆ to find the total genus polynomialof the bouquets Bn, n ≥ 1 in [31], which are defined as follows.

Denotes by zθ and z−1θ = 1/zθ the multivariate monomialsk

i=1zθi and 1/

ki=1

zθi ,

where θ = (θ1, θ2, · · · , θk) ⊢ 2n. Then the linear operators Γ, Θ and ∆ are definedby

Γ(z±1θ

) =k

j=1

θj

l=0

θ j

(

z1+lzθj+1−l

zθj

)zθ

±1

Θ(z±1θ ) =k

j=1

(θ2 j + θ j )(zθj+2zθ

zθj

)−1

and

∆(z±1θ ) =

1≤i<j≤k

2θiθ j [(zθj+θi+2

zθjzθi

)zθ±1 + (zθj+θi+2

zθjzθi

)zθ−1]

Denotes by i[Bn](z j ) the sum of all monomial zθ or 1/zθ taken over all embeddingsof Bn into an orientable or non-orientable surface, that is

i[Bn](z j) =

θ⊢2n

iθ(Bn)zθ +

θ⊢2n

iθ(Bn)z−1θ ,

where, iθ(Bn) and iθ(Bn) denote the number of embeddings of Bn into orientableand non-orientable surface of region type θ. They proved in [31] that

i[Bn+1](z j ) = (Γ + Θ + ∆)i[Bn](z j) = (Γ + Θ + ∆)n(1

z2+ z21).

and

G[Bn+1](x) = (Γ + Θ + ∆)n

(1

z2 + z21)|zj=x f or j≥1 and (C ∗).

Where, (C ∗) denotes the condition

(C ∗): replacing the power 1 + n − 2i of x by i if i ≥ 0 and −(1 + n + i) by −i if i ≤ 0.

Since

|Aut 12

Bn|2ε(Bn)

=2nn!

2n= 2n−1(n − 1)!

and Γ, Θ, ∆ are linear, by the Corollary 1.4.3 we know that




R[Bn+1](x) =(Γ + Θ + ∆)i[Bn](z j)

2nn!|zj=x f or j≥1 and (C ∗)

=(Γ + Θ + ∆)n( 1

z2+ z21)

nk=1

2kk!|zj=x f or j≥1 and (C ∗).

For example, calculation shows that

R[B1](x) = x +1

x;

R[B2](x) = 2 + x +5

x+

4

x2;

R[B3](x) =41

x3+

42

x2+

22

x+ 5 + 10x;

and

R[B4](x) =488

x4+

690

x3+

304

x2+

93

x+ 14 + 70x + 21x2.



Chapter2

On the Automorphisms of a Klein Surface and as-Manifold

Many papers concerned the automorphisms of a Klein surface, such as,[1], [15],[26], [38] for a Riemann surface by using Fuchsian group and [9] − [10], [21] for aKlein surface without boundary by using NEC groups. Since maps is a naturalmodel for the Klein surfaces, an even more efficient approach is, perhaps, by usingthe combinatorial map theory. Establishing some classical results again and findingtheir combinatorial refinement are the central topics in this chapter.

§1. An algebraic definition of a voltage map

1.1 Coverings of a map

For two maps M = (X α,β , P ) and M = (X α,β , P ), call the map M covering the

map M if there is a mapping π : X α,β → X α,β such that ∀x ∈ X α,β ,

απ(x) = πα(x), βπ(x) = πβ (x) and π P (x) = P π(x).

The mapping π is called a covering mapping . For ∀x ∈ X α,β , define the quadricell

set π

−1

(x) by

π−1(x) = x|x ∈ (X α,β and π(x) = x.


Lemma 2.1.1 Let π : X α,β → X α,β be a covering mapping. Then for any twoquadricells x1, x2 ∈ X α,β ,

( i) |π−1(x1)| = |π−1(x2)|.( ii) If x1 = x2, then π−1(x1)

π−1(x2) = ∅.

Proof (i) By the definition of a map, for x1, x2 ∈ X α,β , there exists an element

σ ∈ ΨJ =< α, β, P > such that x2 = σ(x1).Since π is an covering mapping from M to M , it is commutative with α, β and

P . Whence, π is also commutative with σ. Therefore,

π−1(x2) = π−1(σ(x1)) = σ(π−1(x1)).

Notice that σ ∈ ΨJ is an 1 − 1 mapping on X α,β . Hence, |π−1(x1)| = |π−1(x2)|.(ii) If x1 = x2 and there exists an element y ∈ π−1(x1)

π−1(x2), then there

must be x1 = π(y) = x2. Contradicts the assumption. The relation of a covering mapping with an isomorphism is in the following

theorem.



Chapter 2 On the Automorphisms of a Klein Surface and a s-Manifold 27

Theorem 2.1.1 Let π : X α,β → X α,β be a covering mapping. Then π is an isomor-

phism iff π is an 1 − 1 mapping.

Proof If π is an isomorphism between the maps M = (X α,β , P ) and M =(X α,β , P ), then it must be an 1 − 1 mapping by the definition, and vice via.

A covering mapping π from M to M naturally induces a mapping π∗ by thefollowing condition:

∀x ∈ X α,β , g ∈ AutM, π∗ : g → πgπ−1(x).


Theorem 2.1.2 If π : X α,β → X α,β is a covering mapping, then the induced mapping π∗ is a homomorphism from AutM to AutM .

Proof First, we prove that for ∀g ∈ AutM and x ∈ X α,β , π∗(g) ∈ AutM.

Notice that for ∀g ∈ AutM and x ∈ X α,β ,

πgπ−1(x) = π(gπ−1(x)) ∈ X α,β

and ∀x1, x2 ∈ X α,β , if x1 = x2, then πgπ−1(x1) = πgπ−1(x2). Otherwise, assumethat

πgπ−1(x1) = πgπ−1(x2) = x0

∈ X α,β ,

then we have that x1 = πg−1π−1(x0) = x2. Contradicts to the assumption.By the definition, for x ∈ X α,β we get that

π∗α(x) = πgπ−1α(x) = πgαπ−1(x) = παgπ−1(x) = απgπ−1(x) = απ∗(x),

π∗β (x) = πgπ−1β (x) = πgβπ−1(x) = πβgπ−1(x) = βπgπ−1(x) = βπ∗(x).

Notice that π( P ) = P . We get that

π∗P (x) = πgπ−1P (x) = πg P π−1(x) = π P gπ−1(x) = P πgπ−1(x) = P π∗(x).

Therefore, we get that πgπ−1 ∈ AutM , i.e., π∗ : AutM → AutM .Now we prove that π∗ is a homomorphism from Aut M to AutM . In fact, for

∀g1, g2 ∈ AutM , we have that

π∗(g1g2) = π(g1g2)π−1 = (πg1π−1)(πg2π−1) = π∗(g1)π∗(g2).

Whence, π∗ : AutM

→AutM is a homomorphism.




1.2 Voltage maps

For creating a homomorphism between Klein surfaces, voltage maps are exten-sively used, which is introduced by Gustin in 1963 and extensively used by Youngsin 1960s for proving the Heawood map coloring theorem and generalized by Grossin 1974 ([22]). Now it already become a powerful approach for getting regular mapson a surface, see [5], [7], [56]− [57], [65]], especially, [56]− [57]. It often appears as anembedded voltage graph in references. Notice that by using the voltage graph the-ory, the 2-factorable graphs are enumerated in [51]. Now we give a purely algebraicdefinition for voltage maps, not using geometrical intuition and establish its theoryin this section and the next section again.

Definition 2.1.1 Let M = (X α,β , P ) be a map and G a finite group. Call a pair (M, ϑ) a voltage map with group G if ϑ : X α,β → G, satisfying the following condi-tion:

(i) ∀x ∈ X α,β , ϑ(αx) = ϑ(x), ϑ(αβx) = ϑ(βx) = ϑ−1(x);(ii) ∀F = (x,y, · · · , z)(βz, · · · ,β y,β x) ∈ F (M ), the face set of M , ϑ(F ) =

ϑ(x)ϑ(y) · · · ϑ(z) and < ϑ(F )|F ∈ F (u), u ∈ V (M ) >= G, where, F (u) denotes all the faces incident with the vertex u.

For a given voltage graph (M, ϑ), define

X αϑ

,β ϑ =

X α,β ×G

P ϑ =

(x,y,···,z)(αz,···,αy,αx)∈V (M )

g∈G

(xg, yg, · · · , zg)(αzg, · · · , αyg, αxg),

and

αϑ = α

β ϑ =

x∈X α,β ,g∈G

(xg, (βx)gϑ(x)).

where, we use ug denoting an element (u, g) ∈ X α,β × G.It can be shown that M ϑ = (X αϑ,β ϑ, P ϑ) also satisfying the conditions of a map

with the same orientation as the map M . Hence, we can define the lifting map of avoltage map as follows.

Definition 2.1.2 For a voltage map (M, ϑ) with group G, the map M ϑ = (X ϑα,β , P ϑ)is called its lifting map.

For a vertex v = (C )(αCα−1) ∈ V (M ), denote by C the quadricells in thecycle C . The following numerical result is obvious by the definition of a lifting map.

Lemma 2.1.2 The numbers of vertices and edges in the lifting map M ϑ are respective




ν (M ϑ) = ν (M )|G| and ε(M ϑ) = ε(M )|G|

Lemma 2.1.3 Let F = (C ∗)(αC ∗α−1) be a face in the map M . Then there are|G|/o(F ) faces in the lifting map M ϑ with length |F |o(F ) lifted from the face F ,where o(F ) denotes the order of

x∈C

ϑ(x) in the group G.

Proof Let F = (u, v · · · , w)(βw, · · · ,β v ,β u) be a face in the map M and k is thelength of F . Then, by the definition, for ∀g ∈ G, the conjugate cycles

(C ∗

)ϑ

= (ug, vgϑ(u), · · · , ugϑ(F ), vgϑ(F )ϑ(u), · · · , wgϑ(F )2, · · · , wgϑo(F )−1(F ))β (ug, vgϑ(u), · · · , ugϑ(F ), vgϑ(F )ϑ(u), · · · , wgϑ(F )2 , · · · , wgϑo(F )−1(F ))

−1β −1.

is a face in M ϑ with length ko(F ). Therefore, there are |G|/o(F ) faces in the liftingmap M ϑ. altogether. .

Therefore,we get that

Theorem 2.1.3 The Euler characteristic χ(M ϑ) of the lifting map M ϑ of the voltagemap (M, G) is

χ(M ϑ) =

|G

|(χ(M ) + m∈O(F (M ))

(

−1 +

1

m

)),

where O(F (M )) denotes the order o(F ) set of the faces in M .

Proof According to the Lemma 2.1.2 and 2.1.3, the lifting map M ϑ has |G|ν (M )vertices, |G|ε(M ) edges and |G|

m∈O(F (M ))

1m faces. Therefore, we get that

χ(M ϑ) = ν (M ϑ) − ε(M ϑ) + φ(M ϑ)

= |G|ν (M ) − |G|ε(M ) + |G| m∈O(F (M ))

1

m

= |G|(χ(M ) − φ(M ) + m∈O(F (M ))

1

m)

= |G|(χ(M ) +

m∈O(F (M ))

(−1 +1

m)).

§2. Combinatorial conditions for a group being that of a map

Locally characterizing that an automorphism of a voltage map is that of itslifting is well-done in the references [40] − [41]. Among them, a typical result is thefollowing:




An automorphism ζ of a map M with voltage assignment ϑ→

G is an automor-phism of its lifting map M ϑ if for each face F with ϑ(F ) = 1G, ϑ(ζ (F )) = 1G.

Since the central topic in this chapter is found what a finite group is an automor-phism group of a map, i.e., a global question, the idea used in the references [40] −[41]are not applicable. New approach should be used.

2.1 Combinatorial conditions for an automorphism group of a map

First, we characterize an automorphism group of a map.A permutation group G action on Ω is called fixed-free if Gx = 1G for ∀x ∈ Ω.

We have the following.

Lemma 2.2.1 Any automorphism group G of a map M = (X α,β , P ) is fixed-free on X α,β .

Proof For ∀x ∈ X α,β ,since G AutM , we get that Gx (AutM )x. Notice that(AutM )x = 1G. Whence, we know that Gx = 1G, i.e., G is fixed-free.

Notice that the automorphism group of a lifting map has a obvious subgroup,determined by the following lemma.

Lemma 2.2.2 Let M ϑ be a lifting map by the voltage assignment ϑ : X α,β → G.Then G is isomorphic to a fixed-free subgroup of AutM ϑ on V (M ϑ).

Proof For ∀g ∈ G, we prove that the induced action g∗

: X αϑ,β ϑ → X αϑ,β ϑ byg∗ : xh → xgh is an automorphism of the map M ϑ.

In fact, g∗ is a mapping on X αϑ,β ϑ and for ∀xu ∈ X αϑ,β ϑ, we get g∗ : xg−1u → xu.Now if for xh, yf ∈ X αϑ,β ϑ, xh = yf , we have that g∗(xh) = g∗(yf ), that is,

xgh = ygf , by the definition, we must have that x = y and gh = gf , i.e., h = f .Whence, xh = yf , contradicts to the assumption. Therefore, g∗ is 1 − 1 on X αϑ,β ϑ.

We prove that for xu ∈ X αϑ,β ϑ, g∗ ia commutative with αϑ, β ϑ and P ϑ. Noticethat

g∗αϑxu = g∗(αx)u = (αx)gu = αxgu = αg∗(xu);

g∗β ϑ(xu) = g∗(βx)uϑ(x) = (βx)guϑ(x) = βxguϑ(x) = β ϑ(xgu) = β ϑg∗(xu)

and

g∗P ϑ(xu)

= g∗

(x,y,···,z)(αz,···,αy,αx)∈V (M )

u∈G

(xu, yu, · · · , zu)(αzu, · · · , αyu, αxu)(xu)

= g∗yu = ygu




= (x,y,···,z)(αz,···,αy,αx)∈V (M ) gu∈G

(xgu

, ygu

,· · ·

, zgu

)(αzgu

,· · ·

, αygu

, αxgu

)(xgu

)

= P ϑ(xgu) = P ϑg∗(xu).

Therefore, g∗ is an automorphism of the lifting map M ϑ.To see g∗ is fixed-free on V (M ), choose ∀u = (xh, yh, · · · , zh)(αzh, · · · , αyh, αxh) ∈

V (M ), h ∈ G. If g∗(u) = u, i.e.,

(xgh, ygh, · · · , zgh)(αzgh, · · · , αygh, αxgh) = (xh, yh, · · · , zh)(αzh, · · · , αyh, αxh).

Assume that xgh = wh, where wh

∈ xh, yh,

· · ·, zh, αxh, αyh,

· · ·, αzh

. By the

definition, there must be that x = w and gh = h. Therefore, g = 1G, i.e., ∀g ∈ G,g∗ is fixed-free on V (M ).

Now define τ : g∗ → g. Then τ is an isomorphism between the action of elementsin G on X αϑ,β ϑ and the group G itself.

According to the Lemma 2.2.1, given a map M and a group G AutM , we candefine a quotient map M/G = (X α,β /G, P /G) as follows.

X α,β /G = xG|x ∈ X α,β ,

where xG denotes an orbit of G action on X α,β containing x and

P /G = (x,y,···,z)(αz,···,αy,αx)∈V (M )

(xG, yG, · · ·)(· · · , αyG, αxG),

since G action on X α,β is fixed-free.Notice that the map M may be not a regular covering of its quotient map M/G.

We have the following theorem characterizing a fixed-free automorphism group of amap on V (M ).

Theorem 2.2.1 An finite group G is a fixed-free automorphism group of a mapM = (X α,β , P ) on V (M ) iff there is a voltage map (M/G,G) with an assignment ϑ : X α,β /G → G such that M ∼= (M/G)ϑ.

Proof The necessity of the condition is already proved in the Lemma 2 .2.2. Weonly need to prove its sufficiency.

Denote by π : M → M/G the quotient map from M to M/G. For each elementof π−1(xG), we give it a label. Choose x ∈ π−1(xG). Assign its label l : x → x1G ,i.e., l(x) = x1G . Since the group G acting on X α,β is fixed-free, if u ∈ π−1(xG) andu = g(x), g ∈ G, we label u with l(u) = xg. Whence, each element in π−1(xG) islabelled by a unique element in G.

Now we assign voltages on the quotient map M/G = (X α,β /G, P /G). If βx =y, y ∈ π−1(yG) and the label of y is l(y) = y∗h, h ∈ G, where, l(y∗) = 1G, thenwe assign a voltage h on xG,i.e., ϑ(xG) = h. We should prove this kind of voltageassignment is well-done, which means that we must prove that for

∀v

∈π−1(xG)




Corollary 2.2.2 If M is a non-orientable map of genus q, G

AutM is fixed-freeon V (M ) and the quotient map M/G with genus δ, then

|G| =q − 2

δ − 2 +

m∈O(F (M/G))(1 − 1

m))

.

Particularly, if M/G is projective planar, then

|G| =q − 2

−1 +

m∈O(F (M/G))(1 − 1

m))

.

By applying the Theorem 2.2.1, we can also calculate the Euler characteristicof the quotient map, which enables us to get the following result for a group beingthat of a map.

Theorem 2.2.3 If a group G, G AutM , then

χ(M ) +

g∈G,g=1G

(|Φv(g)| + |Φf (g)|) = |G|χ(M/G),

where, Φv(g) = v|v ∈ V (M ), vg = v and Φf (g) = f |f ∈ F (M ), f g = f , and if G is fixed-free on V (M ), then

χ(M ) + g∈G,g=1G

|Φf (g)| = |G|χ(M/G).

Proof By the definition of a quotient map, we know that

φv(M/G) = orbv(G) =1

|G|g∈G

|Φv(g)|

and

φf (M/G) = orbf (G) =1

|G

| g∈G

|Φf (g)|,

by applying the Burnside Lemma. Since G is fixed-free on X α,β by the Lemma 2.1,we also know that

ε(M/G) =ε(M )

|G| .

Applying the Euler-Poincare formula for the quotient map M/G, we get thatg∈G

|Φv(g)||G| − ε(M )

|G| +

g∈G

|Φf (g)||G| = χ(M/G).




Whence, we have

g∈G

|Φv(g)| − ε(M ) +g∈G

|Φf (g)| = |G|χ(M/G).

Notice that ν (M ) = |Φv(1G)|, φ(M ) = |Φf (1G)| and ν (M ) − ε(M ) + φ(M ) = χ(M ).We know that

χ(M ) +

g∈G,g=1G

(|Φv(g)| + |Φf (g)|) = |G|χ(M/G).

Now if G is fixed-free on V (M ), by the Theorem 2.1, there is a voltage assignmentϑ on the quotient map M/G such that M

∼= (M/G)ϑ. According to the Lemma

2.1.2, we know that

ν (M/G) =ν (M )

|G| .

Whence,

g∈G|Φv(g)| = ν (M ) and

g∈G,g=1G

(|Φv(g)| = 0. Therefore, we get that

χ(M ) +

g∈G,g=1G

|Φf (g)| = |G|χ(M/G).

Consider the properties of the group G on F (M ), we get the following interestingresults.

Corollary 2.2.3 If a finite group G, G AutM is fixed-free on V (M ) and transitiveon F (M ), for example, M is regular and G = AutM , then M/G is an one face mapand

χ(M ) = |G|(χ(M/G) − 1) + φ(M )

Particularly, for an one face map, we know that

Corollary 2.2.4 For an one face map M , if G, G AutM is fixed-free on V (M ),then

χ(M ) − 1 = |G|(χ(M/G) − 1),

and |G|, especially, |AutM | is an integer factor of χ(M ) − 1.

Remark 2.2.1 For an one face planar map, i.e., the plane tree, the only fixed-freeautomorphism group on its vertices is the trivial group by the Corollary 2.4.

2.2 The measures on a map

On the classical geometry, its central question is to determine the measures onthe objects, such as the distance, angle, area, volume, curvature, . . .. For maps being




a combinatorial model of Klein surfaces, we also wish to introduce various measureson a map and enlarge its application filed to other branch of mathematics..

2.2.1 The angle on a map

For a map M = (X α,β , P ), x ∈ X α,β , the permutation pair (x, P x), (αx, P −1αx)is called an angle incident with x, which is introduced by Tutte in [66]. We provein this section that any automorphism of a map is a conformal mapping and affirmthe Theorem 1.2.7 in Chapter 1 again.

Define an angle transformation Θ of angles of a map M = (X α,β , P ) as follows.

Θ =

x∈X α,β

(x, P x).

Then we have

Theorem 2.2.4 Any automorphism of a map M = (X α,β , P ) is conformal.

Proof By the definition, for ∀g ∈ AutM , we know that

αg = gα, βg = gβ and P g = gP .Therefore, for ∀x ∈ X α,β , we have

Θg(x) = (g(x), P g(x))

and

gΘ(x) = g(x, P x) = (g(x), P g(x)).

Whence, we get that for ∀x ∈ X α,β , Θg(x) = gΘ(x). Therefore, we get thatΘg = gΘ,i.e., gΘg−1 = Θ.

Since for ∀x ∈ X α,β , gΘg−1(x) = (g(x), P g(x)) and Θ(x) = (x, P (x)), we havethat

(g(x), P g(x)) = (x, P (x)).

That is, g is a conformal mapping.

2.2.2 The non-Euclid area on a map

For a given voltage map (M, G), its non-Euclid area µ(M, G) is defined by

µ(M, G) = 2π(−χ(M ) +

m∈O(F (M ))

(−1 +1

m)).

Particularly, since any map M can be viewed as a voltage map (M, 1G), we get thenon-Euclid area of a map M

µ(M ) = µ(M, 1G

) =−

2πχ(M ).




Notice that the area of a map is only dependent on the genus of the surface. Weknow the following result.

Theorem 2.2.5 Two maps on one surface have the same non-Euclid area.

By the non-Euclid area, we get the Riemann-Hurwitz formula in Klein surfacetheory for a map in the following result.

Theorem 2.2.6 If G AutM is fixed-free on V (M ), then

|G| =µ(M )

µ(M/G,ϑ),

where ϑ is constructed in the proof of the Theorem 2.2.1.

Proof According to the Theorem 2.2.2, we know that

|G| =−χ(M )

−χ(M ) +

m∈O(F (M ))(−1 + 1

m )

=−2πχ(M )

2π(−χ(M ) +

m∈O(F (M ))(−1 + 1

m ))=

µ(M )

µ(M/G,ϑ).

As an interesting result, we can obtain the same result for the non-Euclid areaof a triangle as the classical differential geometry.

Theorem 2.2.7([42]) The non-Euclid area µ(∆) of a triangle ∆ on a surface S with internal angles η ,θ ,σ is

µ(∆) = η + θ + σ − π.

Proof According to the Theorem 1.2.1 and 2.2.5, we can assume there is a trian-gulation M with internal angles η ,θ ,σ on S and with an equal non-Euclid area oneach triangular disk. Then

φ(M )µ(∆) = µ(M ) = −2πχ(M )

= −2π(ν (M ) − ε(M ) + φ(M )).

Since M is a triangulation, we know that

2ε(M ) = 3φ(M ).

Notice that the sum of all the angles in the triangles on the surface S is 2πν (M ),we get that




φ(M )µ(∆) = −2π(ν (M ) − ε(M ) + φ(M )) = (2ν (M ) − φ(M ))π

=φ(M )

i=1

[(η + θ + σ) − π] = φ(M )(η + θ + σ − π).

Whence, we get that

µ(∆) = η + θ + σ − π.

§3. A combinatorial refinement of Huriwtz theorem

In 1893, Hurwitz obtained a famous result for the orientation-preserving auto-morphism group Aut+S of a Riemann surface S ([11][18][22]):

For a Riemann surface S of genus g(S ) ≥ 2, Aut+S ≤ 84(g(S ) − 1).

We have known that the maps are the combinatorial model for Klein surfaces,especially, the Riemann surfaces. What is its combinatorial counterpart? What wecan say for the automorphisms of a map?

For a given graph Γ, define a graphical property P to be its a kind of subgraphs,

such as, regular subgraphs, circuits, trees, stars, wheels, · · ·. Let M = (X α,β , P ) bea map. Call a subset A of X α,β has the graphical property P if the underlying graphof A has property P . Denote by A(P, M ) the set of all the A subset with propertyP in the map M .

For refinement the Huriwtz theorem, we get a general combinatorial result in thefollowing.

Theorem 2.3.1 Let M = (X α,β , P ) be a map. Then for ∀G AutM ,

[|vG||v ∈ V (M )] | |G|

and

|G| | |A||A(P, M )|,where[a,b, · · ·] denotes least common multiple of a,b, · · ·.

Proof According to a well-known result in the permutation group theory, for∀v ∈ V (M ), we know |G| = |Gv||vG|. Therefore, |vG| | |G|. Whence,

[|vG||v ∈ V (M )] | |G|.The group G is fixed-free action on X α,β , i.e., ∀x ∈ X α,β , we have |Gx| = 1 (see

also [28]).




Now we consider the action of the automorphism group G onA

(P, M ). Noticethat if A ∈ A(P, M ), then then ∀g ∈ G,Ag) ∈ A(P, M ), i.e., AG ⊆ A(P, M ). Thatis, the action of G on A(P, M ) is closed. Whence, we can classify the elements inA(P, M ) by G. For ∀x, y ∈ A(P, M ), define x ∼ y if and only if there is an elementg, g ∈ G such that xg = y.

Since |Gx| = 1, i.e., |xG| = |G|, we know that each orbit of G action on X α,β hasa same length |G|. By the previous discussion, the action of G on A(P, M ) is closed,therefore, the length of each orbit of G action on A(P, M ) is also |G|. Notice thatthere are |A||A(P, M )| quadricells in A(P, M ). We get that

|G| | |A||A(P, M )|.This completes the proof.

Choose property P to be tours with each edge appearing at most 2 in the mapM . Then we get the following results by the Theorem 2.3.1.

Corollary 2.3.1 Let T r2 be the set of tours with each edge appearing 2 times. Then for ∀G AutM ,

|G| | (l|T r2|, l = |T | =|T |2

≥ 1, T ∈ T r2, ).

Let T r1 be the set of tours without repeat edges. Then

|G| | (2l|T r1|, l = |T | = |T |2

≥ 1, T ∈ T r1, ).

Particularly, denote by φ(i, j) the number of faces in M with facial length i and singular edges j, then

|G| | ((2i − j)φ(i, j), i , j ≥ 1),

where,(a,b, · · ·) denotes the greatest common divisor of a,b, · · ·.Corollary 2.3.2 Let T be the set of trees in the map M . Then for ∀G AutM ,

|G| | (2ltl, l ≥ 1),

where tl denotes the number of trees with l edges.

Corollary 2.3.3 Let vi be the number of vertices with valence i. Then for ∀G AutM ,

|G| | (2ivi, i ≥ 1).

Theorem 2.3.1 is a combinatorial refinement of the Hurwitz theorem. Applyingit, we can get the automorphism group of a map as follows.




Theorem 2.3.2 Let M be an orientable map of genus g(M )≥

2. Then for ∀

GAut+M ,

|G| ≤ 84(g(M ) − 1)

and for ∀G AutM ,

|G| ≤ 168(g(M ) − 1).

Proof Define the average vertex valence ν (M ) and the average face valence φ(M )of a map M by

ν (M ) =1

ν (M )

i≥1

iν i,

φ(M ) =1

φ(M )

j≥1

jφ j,

where,ν (M ),φ(M ),φ(M ) and φ j denote the number of vertices, faces, vertices of valence i and faces of valence j, respectively.

Then we know that ν (M )ν (M ) = φ(M )φ(M ) = 2ε(M ). Whence, ν (M ) = 2ε(M )

ν (M )

and φ(M ) = 2ε(M )

φ(M ). According to the Euler formula, we have that

ν (M ) − ε(M ) + φ(M ) = 2 − 2g(M ),

where,ε(M ), g(M ) denote the number of edges and genus of the map M . We getthat

ε(M ) =2(g(M ) − 1)

(1 − 2

ν (M )− 2

φ(M ))

.

Choose the integers k = ⌈ν (M )⌉ and l = ⌈φ(M )⌉. We have that

ε(M ) ≤2(g(M )

−1)

(1 − 2k − 2l ) .

Because 1− 2k− 2

l > 0, So k ≥ 3, l > 2kk−2

. Calculation shows that the minimum

value of 1 − 2k

− 2l

is 121

and attains the minimum value if and only if (k, l) = (3, 7)or (7, 3). Therefore,

ε(M ≤ 42(g(M ) − 1)).

According to the Theorem 2.3.1 and its corollaries, we know that |G| ≤ 4ε(M )and if G is orientation-preserving, then |G| ≤ 2ε(M ). Whence,

|G

| ≤168(g(M )

−1))




and if G is orientation-preserving, then

|G| ≤ 84(g(M ) − 1)),

with equality hold if and only if G = AutM, (k, l) = (3, 7) or (7, 3) For the automorphism of a Riemann surface, we have

Corollary 2.3.4 For any Riemann surface S of genus g ≥ 2,

4g(S ) + 2 ≤ |Aut+S| ≤ 84(g(S ) − 1)

and

8g(S ) + 4 ≤ |AutS| ≤ 168(g(S ) − 1).

Proof By the Theorem 1.2.6 and 2.3.2, we know the upper bound for |AutS| and|Aut+S|. Now we prove the lower bound. We construct a regular map M k = (X k, P k)on a Riemann surface of genus g ≥ 2 as follows, where k = 2g + 1.

X k = x1, x2, · · · , xk, αx1, αx2, · · · , αxk, βx1, βx2, · · · , βxk,αβx1,αβx2, · · · ,αβxk

P k = (x1, x2, · · · , xk,αβx1,αβx2, · · · ,αβxk)(βxk, · · · , βx2, βx1, αxk, · · · , αx2, αx1).

It can be shown that M k is a regular map, and its orientation-preserving auto-morphism group Aut+M k =< P k >. Direct calculation shows that if k ≡ 0(mod2),M k has 2 faces, and if k ≡ 1, M k is an one face map. Therefore, according to theTheorem 1.2.6, we get that

|Aut+S| ≥ 2ε(M k) ≥ 4g + 2,

and

|AutS| ≥ 4ε(M k) ≥ 8g + 4.

For the non-orientable case, we can also get the bound for the automorphismgroup of a map.

Theorem 2.3.3 Let M be a non-orientable map of genus g′(M ) ≥ 3. Then for ∀G Aut+M ,

|G| ≤ 42(g′(M ) − 2)

and for ∀G AutM ,




|G| ≤ 84(g′(M ) − 2),

with the equality hold iff M is a regular map with vertex valence 3 and face valence7 or vice via.

Proof Similar to the proof of the Theorem 2.3.2, we can also get that

ε(M ≤ 21(g′(M ) − 2))

and with equality hold if and only if G = AutM and M is a regular map with vertexvalence 3, face valence 7 or vice via. According to the Corollary 2.3.3, we get that

|G| ≤ 4ε(M )

and if G is orientation-preserving, then

|G| ≤ 2ε(M ).

Whence, for ∀G Aut+M ,

|G| ≤ 42(g′(M ) − 2)

and for ∀G AutM ,

|G| ≤ 84(g′(M ) − 2),

with the equality hold iff M is a regular map with vertex valence 3 and face valence7 or vice via.

Similar to the Hurwtiz theorem for a Riemann surface, we can get the upperbound for a Klein surface underlying a non-orientable surface.

Corollary 2.3.5 For any Klein surface K underlying a non-orientable surface of genus q ≥ 3,

|Aut+K| ≤ 42(q − 2)

and

|AutK| ≤ 84(q − 2).

According to the Theorem 1.2.8, similar to the proof of the Theorem 2.3.2 and2.3.3, we get the following result for the automorphisms of an s-manifold as follows.

Theorem 2.3.4 Let C(T, n) be a closed s-manifold with negative Euler characteristic.Then |AutC(T, n)| ≤ 6n and

|AutC(T, n)| ≤ −21χ(C(T, n)),




with equality hold only if C

(T, n) is hyperbolic, where χ(C

(T, n)) denotes the genusof C(T, n).

Proof The inequality |AutC(T, n)| ≤ 6n is known by the Corollary 2.3.1. Similarto the proof of the Theorem 2.3.2, we know that

ε(C(T, n)) =−χ(C(T, n))

13− 2

k

,

where k = 1ν (C(T,n))

i≥1

iν i ≤ 7 and with the equality holds only if k = 7, i.e., C(T, n)

is hyperbolic.

§4. The order of an automorphism of a Klein surface

Harvey [26] in 1966, Singerman [60] in 1971 and Bujalance [9] in 1983 consideredthe order of an automorphism of a Riemann surface of genus p ≥ 2 and a compactnon-orientable Klein surface without boundary of genus q ≥ 3. Their approach isby using the Fuchsian groups and NEC groups for Klein surfaces. The central ideais by applying the Riemann-Hurwitz equation , stated as follows:

Let G be an NEC graph and G′ be a subgroup of G with finite index. Then

µ(G′

)µ(G)

= [G : G′],

where, µ(G) denotes the non-Euclid area of the group G, which is defined as if

σ = (g; ±; [m1, · · · , mr]; (n11,···,n1s1), · · · , (nk1, · · · , nks))

is the signature of the group G, then

µ(G) = 2π[ηg + k − 2 +r

i=1

(1 − 1/mi) + 1/2k

i=1

si j=1

(1 − 1/nij )],

where, η = 2 if sign(σ) = + and η = 1 otherwise.Notice that we have introduced the conception of non-Euclid area for the voltage

maps and have gotten the Riemann-Hurwitz equation in the Theorem 2 .2.6 for afixed-free on V (M ) group. Similarly, we can find the minimum genus of a map,fixed-free on its vertex set by the voltage assignment on its quotient map and themaximum order of an automorphism of a map.

4.1 The minimum genus of a fixed-free automorphism

Lemma 2.4.1 Let N =k

i=1 pri

i , p1 < p2 < · · · < pk be the arithmetic decomposition

of the integer N and mi ≥

1, mi|

N for i = 1, 2,· · ·

, k. Then for any integer s≥

1,




si=1

(1 − 1mi

) ≥ 2(1 − 1 p1

)⌊s2⌋.

Proof If s ≡ 0(mod2), it is obvious that

si=1

(1 − 1

mi) ≥

si=1

(1 − 1

p1) ≥ (1 − 1

p1)s.

Now assume that s ≡ 1(mod2) and there are mij = p1, j = 1, 2, · · · , l. If theassertion is not true, we must have that

(1 − 1 p1

)(l − 1) >l

j=1

(1 − 1mij

) ≥ (1 − 1 p2

)l.

Whence, we get that

(1 − 1

p1)l > (1 − 1

p2)l + 1 − 1

p1> (1 − 1

p1)l.

A contradiction. Therefore, we have that

si=1

(1 − 1

mi) ≥ 2(1 − 1

p1)⌊s

2⌋.

Lemma 2.4.2 For a map M = (X α,β , P ) with φ(M ) faces and N =k

i=1 pri

i , p1 <

p2 < · · · < pk, the arithmetic decomposition of the integer N , there exists a voltageassignment ϑ : X α,β → Z N such that for ∀F ∈ F (M ), o(F ) = p1 if φ(M ) ≡ 0(mod2)or there exists a face F 0 ∈ F (M ), o(F ) = p1 for ∀F ∈ F (M ) \F 0, but o(F 0) = 1.

Proof Assume that f 1, f 2, · · · , f n, where,n = φ(M ), are the n faces of the mapM . By the definition of voltage assignment, if x,βx or x,αβx appear on one facef i, 1 ≤ i ≤ n altogether, then they contribute to ϑ(f i) only with ϑ(x)ϑ−1(x) = 1Z N .Whence, not loss of generality, we only need to consider the voltage xij on the

common boundary among the face f i and f j for 1 ≤ i, j ≤ n. Then the voltageassignment on the n faces are

ϑ(f 1) = x12x13 · · · x1n,

ϑ(f 2) = x21x23 · · · x2n,

· · · · · · · · · · · · · · · · · ·

ϑ(f n

) = xn1

xn2 · · ·

xn(n−1)

.




We wish to find an assignment on M which can enables us to get as manyfaces as possible with the voltage of order p1. Not loss of generality, we can chooseϑ p1(f 1) = 1Z N in the first. To make ϑ p1(f 2) = 1Z N , choose x23 = x−113 , · · · , x2n = x−11n .If we have gotten ϑ p1(f i) = 1Z N and i < n if n ≡ 0(mod2) or i < n − 1 if n ≡ 1(mod2), we can choose that

x(i+1)(i+2) = x−1i(i+2), x(i+1)(i+3) = x−1i(i+3), · · · , x(i+1)n = x−1in ,

which also make ϑ p1(f i+1) = 1Z N .Now if n ≡ 0(mod2), this voltage assignment makes each face f i, 1 ≤ i ≤ n

satisfying that ϑ p1(f i) = 1Z N . But if n ≡ 1(mod2), it only makes ϑ p1(f i) = 1Z N for1

≤i≤

n−

1, but ϑ(f n) = 1Z N . This completes the proof. Now we can prove a result for the minimum genus of a fixed-free automorphism

of a map.

Theorem 2.4.1 Let M = (X α,β , P ) be a map and N = pr11 · · · prk

k , p1 < p2 <· · · < pk, be the arithmetic decomposition of the integer N . Then for any voltageassignment ϑ : X α,β → Z N ,

(i) if M is orientable, the minimum genus gmin of the lifting map M ϑ which admits an automorphism of order N , fixed-free on V (M ϑ), is

gmin = 1 + N g(M ) − 1 + (1 −m∈O(F (M ))

1

p1)⌊φ(M )

2⌋.

(ii) if M is non-orientable, the minimum genus g′min of the lifting map M ϑ which admits an automorphism of order N , fixed-free on V (M ϑ), is

g′min = 2 + N g(M ) − 2 + 2(1 − 1

p1)⌊φ(M )

2⌋.

Proof (i) According to the Theorem 2.2.1, we know that

2 − 2g(M ϑ) = N (2 − 2g(M )) +

m∈O(F (M ))

(−1 +1

m).

Whence,

2g(M ϑ) = 2 + N 2g(M ) − 2 +

m∈O(F (M ))

(1 − 1

m).

Applying the Lemma 2.4.1 and 2.4.2, we get that

gmin = 1 + N g(M ) − 1 + (1 − 1

p1)⌊φ(M )

2⌋

.(ii) Similarly, by the Theorem 2.2.1, we know that




2 − g(M ϑ) = N (2 − g(M )) + m∈O(F (M ))

(−1 + 1m

).

Whence,

g(M ϑ) = 2 + N g(M ) − 2 +

m∈O(F (M ))

(1 − 1

m).

Applying the Lemma 2.4.1 and 2.4.2, we get that

g′min = 2 + N g(M ) − 2 + 2(1 − 1

p1)⌊φ(M )

2⌋.

4.2 The maximum order of an automorphism of a map

For the maximum order of an automorphism of a map, we have the followingresult.

Theorem 2.4.2 The maximum order N max of an automorphism g of an orientablemap M of genus≥ 2 is

N max ≤ 2g(M ) + 1

and the maximum order N ′max of a non-orientable map of genus≥ 3 is

N ′max ≤ g(M ) + 1,

where g(M ) is the genus of the map M .

Proof According to the Theorem 2.2.3, denote G =< g >, we get that

χ(M ) +

g∈G,g=1G

(|Φv(g)| + |Φf (g)|) = |G|χ(M/G),

where, Φf (g) = F |F ∈ F (M ), F g = F and Φv (g) = v|v ∈ V (M ), vg = v. If

g = 1G, direct calculation shows that Φf (g) = Φf (g2

) and Φv(g) = Φv(g2

). Whence,g∈G,g=1G

|Φv(g)| = (|G| − 1)|Φv(g)|

and

g∈G,g=1G

|Φf (g)| = (|G| − 1)|Φf (g)|.


χ(M ) + (|G

| −1)

|Φ

v(g)

|+ (

|G

| −1)

|Φ

f (g)

|=

|G

|χ(M/G)




Whence, we have that

χ(M ) − (|Φv(g)| + |Φf (g)|) = |G|(χ(M/G) − (|Φv(g)| + |Φf (g)|)).

If χ(M/G) − (|Φv(g)| + |Φf (g)|) = 0,i.e., χ(M/G) = |Φv(g)| + |Φf (g)| ≥ 0, thenwe get that g(M ) ≤ 1 if M is orientable or g(M ) ≤ 2 if M is non-orientable. Con-tradicts to the assumption. Therefore, χ(M/G) − (|Φv(g)| + |Φf (g)|) = 0. Whence,we get that

|G| =χ(M ) − (|Φv(g)| + |Φf (g)|)

χ(M/G) − (|Φv(g)| + |Φf (g)|) = H (v, f ; g).

Notice that |G|, χ(M )− (|Φv(g)| + |Φf (g)|) and χ(M/G)− (|Φv(g)|+ |Φf (g)|) are in-tegers. We know that the function H (v, f ; g) takes its maximum value at χ(M/G)−(|Φv(g)| + |Φf (g)|) = −1 since χ(M ) ≤ −1. But in this case, we get that

|G| = |Φv(g)| + |Φf (g)| − χ(M ) = 1 + χ(M/G) − χ(M ).

We divide the discussion into to cases.

Case 1 M is orientable.

Since χ(M/G) + 1 = (|Φv(g)| + |Φf (g)|) ≥ 0, we know that χ(M/G) ≥ −1.Whence, χ(M/G) = 0 or 2. Therefore, we have that

|G| = 1 + χ(M/G) − χ(M ) ≤ 3 − χ(M ) = 2g(M ) + 1.

That is, N max ≤ 2g(M ) + 1.

Case 2 M is non-orientable.

In this case, since χ(M/G) ≥ −1, we know that χ(M/G) = −1, 0, 1 or 2.Whence, we have that

|G| = 1 + χ(M/G) − χ(M ) ≤ 3 − χ(M ) = g(M ) + 1.


According to this theorem, we get the following result for the order of an auto-morphism of a Klein surface without boundary by the Theorem 1.2.7, which is evenmore better than the results already known.

Corollary 2.4.1 The maximum order of an automorphism of a Riemann surfaceof genus≥ 2 is 2g(M ) + 1, and the maximum order of an automorphism of a non-orientable Klein surface of genus≥ 3 without boundary is g(M ) + 1.

The maximum order of an automorphism of a map can be also determined byits underlying graph, which is stated as follows.




Theorem 2.4.3 Let M be a map underlying the graph G and omax

(M, g), omax

(G, g)be the maximum order of orientation-preserving automorphism in AutM and in Aut1

2G. Then

omax(M, g) ≤ omax(G, g),

and the equality hold for at least one map underlying the graph G.

The proof of the Theorem 2.4.3 will be delayed to the next chapter after we provethe Theorem 3.1.1. By this theorem, we get the following interesting results.

Corollary 2.4.2 The maximum order of an orientation-preserving automorphism of

a complete map Kn, n ≥ 3, is at most n.

Corollary 2.4.3 The maximum order of an orientation-preserving automorphism of a plane tree T is at most | T |− 1 and attains the upper bound only if the underlying tree is the star.



Chapter3

On the Automorphisms of a Graph on Surfaces

For determining the automorphisms of a map, an alternate approach is to con-sider the action of the semi-arc automorphism group of its underlying graph onthe quadricells and to distinguish which is an automorphism of the map and whichis not. This approach is first appeared in the reference [43] as an initial step forthe enumeration of the non-equivalent embeddings of a graph on surfaces, and alsoimportant for enumeration unrooted maps underlying a graph on surfaces used inChapter 4.

§1. A necessary and sufficient condition for a group of a graph beingthat of a map

Let Γ = (V, E ) be a connected graph. Its automorphism is denoted by AutΓ.Choose the base set X = E (Γ). Then its quadricells X α,β is defined to be:

X α,β =

x∈X

x,αx,βx,βαβx,

where, K = 1,α,β ,αβ is the Klein 4- elements group.For ∀g ∈ AutΓ, define an induced action g|X α,β of g on X α,β as follows.

For ∀x ∈ X α,β , if xg = y, then define (αx)g = αy, (βx)g = βy and (αβx)g = αβy.Let M = (X α,β , P ) be a map. According to the Theorem 1.2.5, for an auto-

morphism g ∈ AutM and g|V (M ) : u → v , u ,v ∈ V (M ), if ug = v, then call gan orientation-preserving automorphism . if ug = v−1, then call g an orientation-reversing automorphism . For any g ∈ AutM , it is obvious that g|Γ is orientation-preserving or orientation-reversing and the product of two orientation-preserving au-tomorphisms or orientation-reversing automorphisms is orientation-preserving, theproduct of an orientation-preserving automorphism with an orientation-reversing

automorphism is orientation-reversing .For a subgroup G AutM , define G+ G being the orientation-preservingsubgroup of G. Then the index of G+ in G is 2. Assume the vertex v to bev = (x1, x2, · · · , xρ(v))(αxρ(v), · · · , αx2, αx1). Denote by < v > the cyclic groupgenerated by v. Then we get the following property for the automorphisms of amap.

Lemma 3.1.1 Let G AutM be an automorphism group of a map M . Then ∀v ∈ V (M ),

( i) if ∀g ∈ G,g is orientation-preserving, then Gv < v >, is a cyclic group;( ii) Gv < v > × < α >.



Chapter 3 On the Automorphisms of a Graph on Surfaces 49

Proof (i)Let M = (X α,β

,P

). since for any∀

g∈

G, g is orientation-preserving,we know for ∀v ∈ V (M ), h ∈ Gv, vh = v. Assume the vertex

v = (x1, x2, · · · , xρ(v))(αxρ(v), αxρ(v)−1, · · · , αx1).

Then

[(x1, x2, · · · , xρ(v))(αxρ(v), · · · , αx2, αx1)]h = (x1, x2, · · · , xρ(v))(αxρ(v), · · · , αx2, αx1).

Therefore, if h(x1) = xk+1, 1 ≤ k ≤ ρ(v), then

h = [(x1, x2, · · · , xρ(v))(αxρ(v), αxρ(v)−1, · · · , αx1)]k = vk.

If h(x1) = αxρ(v)−k+1, 1 ≤ k ≤ ρ(v), then

h = [(x1, x2, · · · , xρ(v))(αxρ(v), αxρ(v)−1, · · · , αx1)]kα = vkα.

But if h = vkα, then we know that vh = vα = v−1, i.e., h is not orientation-preserving. Whence, h = vk, 1 ≤ k ≤ ρ(v), i.e., each element in Gv is the power of v. Assume ξ is the least power of elements in Gv. Then Gv =< vξ > < v > is acyclic group generated by vξ.

(ii)For ∀g ∈ Gv, vg = v, i.e.,

[(x1, x2, · · · , xρ)(αxρ, αxρ−1, · · · , αx1)]g = (x1, x2, · · · , xρ)(αxρ, αxρ−1, · · · , αx1).

Similar to the proof of (i), we know there exists an integer s, 1 ≤ s ≤ ρ, suchthat g = vs or g = vsα. Whence, g ∈< v > or g ∈< v > α, i.e.,

Gv < v > × < α > .

Lemma 3.1.2 Let Γ be a connected graph. If G AutΓ, and ∀v ∈ V (Γ),Gv <v > × < α >, then the action of G on X α,β is fixed-free.

Proof Choose a quadricell x ∈ X α,β . We prove that Gx = 1G. In fact, if g ∈ Gx, then xg = x. Particularly, the incident vertex u is stable under the actionof g, i.e., ug = u. assume

u = (x, y1, · · · , yρ(u)−1)(αx,αyρ(u)−1, · · · , αy1),

then since Gu < u > × < α >, we get that

xg = x, yg1 = y1, · · · , yg

ρ(u)−1 = yρ(u)−1

and




(αx)g = αx, (αy1)g = αy1, · · · , (αyρ(u)−1)g = αyρ(u)−1,

that is, for any quadricell eu incident with the vertex u, egu = eu. According to the

definition of the induced action AutΓ on X α,β , we know that

(βx)g = βx, (βy1)g = βy1, · · · , (βyρ(u)−1)g = βyρ(u)−1

and

(αβx)g = αβx, (αβy1)g = αβy1, · · · , (αβyρ(u)−1)g = αβyρ(u)−1.

Whence, for any quadricell y ∈ X α,β , assume the incident vertex of y is w, then by

the connectivity of the graph Γ, we know that there is a path P (u, w) = uv1v2 · · · vswin Γ connecting the vertex u andw. Not loss of generality, we assume that βyk isincident with the vertex v1. Since (βyk)g = βyk and Gv1 < v1 > × < α >, weknow that for any quadricell ev1 incident with the vertex v1, eg

v1 = ev1.Similarly, if a quadricell evi incident with the vertex vi is stable under the action

of g, i.e., (evi)g = evi , then we can prove that any quadricell evi+1 incident with the

vertex vi+1 is stable under the action of g. This process can be well done until wearrive the vertex w. Therefore, we can get that any quadricell ew incident with thevertex w is stable under the action of g. Particularly, we have that yg = y.

Therefore, we get that g = 1G. Whence,Gx = 1G. Now we prove a necessary and sufficient condition for a subgroup of a graph

being an automorphism group of a map underlying this graph.

Theorem 3.1.1 Let Γ be a connected graph. If G AutΓ, then G is an auto-morphism group of a map underlying the graph Γ iff for ∀v ∈ V (Γ), the stabler Gv < v > × < α >.

Proof According to the Lemma 3.1.1(ii), the condition of the Theorem 3.1.1 isnecessary. Now we prove its sufficiency.

By the Lemma 3.1.2, we know that the action of G on X α,β is fixed-free, i.e.,for∀x ∈ X α,β , |Gx| = 1. Whence, the length of orbit of x under the action G is

|xG

|=

|Gx

||xG

|=

|G

|, i.e., for

∀x

∈ X α,β , the length of x under the action of G is

|G|.Assume that there are s orbits O1, O2, · · · , Os of G action on V (Γ), where, O1 =

u1, u2, · · · , uk, O2 = v1, v2, · · · , vl,· · ·,Os = w1, w2, · · · , wt. We construct theconjugatcy permutation pair for every vertex in the graph Γ such that they productP is stable under the action of G.

Notice that for ∀u ∈ V (Γ), since |G| = |Gu||uG|, we know that [k,l, · · · , t]| |G|.In the first, we determine the conjugate permutation pairs for each vertex in the

orbit O1. Choose any vertex u1 ∈ O1, assume that the stabler Gu1 is 1G, g1, g2g1, · · · ,m−1i=1

gm−

where, m = |Gu1| and the quadricells incident with vertex u1 is

N (u1) in the graph

Γ . We arrange the elements in N (u1) as follows.




Choose a quadricell ua

1 ∈ N (u1). We use G

u1

action on ua

1and αua

1, respec-

tively. Then we get the quadricell set A1 = ua1, g1(ua

1), · · · ,m−1i=1

gm−i(ua1) and

αA1 = αua1, αg1(ua

1), · · · , αm−1i=1

gm−i(ua1). By the definition of the action of an

automorphism of a graph on its quadricells we know that A1

αA1 = ∅. Arrange

the elements in A1 as−→A1 = ua

1, g1(ua1), · · · ,

m−1i=1

gm−i(ua1).

If N (u1) \ A1

αA1 = ∅, then the arrangement of elements in N (u1) is−→A1.

If N (u1) \ A1

αA1 = ∅, choose a quadricell ub1 ∈ N (u1) \ A1

αA1. Similarly,

using the group Gu1 acts on ub1, we get that A2 =

ub1, g1(ub

1),

· · ·,

m−1

i=1

gm−i(ub1)

and

αA2 = αub1, αg1(ub

1), · · · , αm−1i=1

gm−i(ub1). Arrange the elements in A1

A2 as

−−−−−→A1

A2 = ua

1, g1(ua1), · · · ,

m−1i=1

gm−i(ua1); ub

1, g1(ub1), · · · ,

m−1i=1

gm−i(ub1).

If N (u1) \ (A1

A2

αA1

αA2) = ∅, then the arrangement of elements in

A1

A2 is−−−−−→A1

A2. Otherwise, N (u1) \ (A1

A2

αA1

αA2) = ∅. We can

choose another quadricell uc1 ∈ N (u1) \ (A1

A2

αA1

αA2). Generally, If we

have gotten the quadricell sets A1, A2, · · · , Ar, 1 ≤ r ≤ 2k, and the arrangement of

element in them is −−−−−−−−−−−−−−→A1A2 · · ·Ar, if N (u1) \ A1 A2 · · ·Ar αA1 αA2 · · ·αAr) = ∅, then we can choose an element ud1 ∈ N (u1)\(A1

A2

· · ·Ar

αA1αA2

· · ·αAr) and define the quadricell set

Ar+1 = ud1, g1(ud

1), · · · ,m−1i=1

gm−i(ud1)

αAr+1 = αud1, αg1(ud

1), · · · , αm−1i=1

gm−i(ud1)

and the arrangement of elements in Ar+1 is

−−→Ar+1 = ud

1, g1(ud1), · · · ,

m−1i=1

gm−i(ud1).

Define the arrangement of elements inr+1

j=1A j to be

−−−→r+1

j=1

A j =

−−−→ri=1

Ai;−−→Ar+1.

Whence,




N (u1) = (k

j=1

A j)

(αk

j=1

A j )

and Ak is obtained by the action of the stabler Gu1 on ue1. At the same time, the

arrangement of elements in the subsetk

j=1A j of N (u1) to be

−−−→k

j=1

A j .

Define the conjugatcy permutation pair of the vertex u1 to be

u1 = (C )(αC −1α),

where

C = (ua1, ub

1, · · · , ue1; g1(ua

1), g1(ub1), · · · , g1(ue

1), · · · ,m−1i=1

(ua1),

m−1i=1

(ub1), · · · ,

m−1i=1

(ue1)).

For any vertex ui ∈ O1, 1 ≤ i ≤ k, assume that h(u1) = ui, where h ∈ G, thenwe define the conjugatcy permutation pair ui of the vertex ui to be

ui = hu1

= (C h)(αC −1α−1).

Since O1 is an orbit of the action G on V (Γ), then we have that

(k

i=1

ui)G =

ki=1

ui .

Similarly, we can define the conjugatcy permutation pairs v1 , v2, · · · , vl, · · · , w1, w2, · · · , wt of vertices in the orbits O2, · · · , Os. We also have that

(l

i=1

vi)G =

li=1

vi .

· · · · · · · · · · · · · · · · · · · · ·

(t

i=1

wi)G =

ti=1

wi.

Now define the permutation

P = (k

i=1

ui) × (l

i=1

vi) × · · · × (t

i=1

wi).

Then since O1, O2, · · · , Os are the orbits of V (Γ) under the action of G, we getthat




P G = (k

i=1

ui)G × (

li=1

vi)G × · · · × (

ti=1

wi)G

= (k

i=1

ui) × (l

i=1

vi) × · · · × (t

i=1

wi) = P .

Whence, if we define the map M = (X α,β , P )then G is an automorphism of themap M .

For the orientation-preserving automorphism, we have the following result.

Theorem 3.1.2 Let Γ be a connected graph. If G AutΓ, then G is an orientation-preserving automorphism group of a map underlying the graph Γ iff for ∀v ∈ V (Γ),the stabler Gv < v > is a cyclic group.

Proof According to the Lemma 3.1.1(i)we know the necessary. Notice the ap-proach of construction the conjugatcy permutation pair in the proof of the Theorem3.1.1. We know that G is also an orientation-preserving automorphism group of themap M .

According to the Theorem 3.1.2, we can prove the Theorem 2.4.3 now.

The Proof of the Theorem 2.4.3

Since every subgroup of a cyclic group is also a cyclic group, we know that anycyclic orientation-preserving automorphism group of the graph Γ is an orientation-preserving automorphism group of a map underlying the graph Γ by the Theorem3.1.2. Whence, we get that

omax(M, g) ≤ omax(G, g).

Corollary 3.1.1 For any positive integer n, there exists a vertex transitive mapM underlying a circultant such that Z n is an orientation-preserving automorphism group of the map M .

Remark 3.1.1 Gardiner et al proved in [20] that if add an additional condition ,i.e, G is transitive on the vertices in Γ, then there is a regular map underlying thegraph Γ.

§2. The automorphisms of a complete graph on surfaces

A map is called a complete map if its underlying graph is a complete graph.For a connected graph Γ, the notations E O(Γ), E N (Γ) and E L(Γ) denote the embed-dings of Γ on the orientable surfaces, non-orientable surfaces and locally surfaces,




respectively. For∀

e = (u, v)∈

E (Γ), its quadricell Ke =

e,αe,βe,αβe

can berepresented by Ke = uv+, uv−, vu+, vu−.

Let K n be a complete graph of order n. Label its vertices by integers 1, 2,...,n.Then its edge set is ij|1 ≤ i, j ≤ n, i = j ij = ji, and

X α,β (K n) = i j+ : 1 ≤ i, j ≤ n, i = ji j− : 1 ≤ i, j ≤ n, i = j,

α =

1≤i,j≤n,i= j

(i j+, i j−),

β =

1≤i,j≤n,i= j

(i j+, i j+)(i j−, i j−).

We determine all the automorphisms of complete maps of order n and give theyconcrete representation in this section.

First, we need some useful lemmas for an automorphism of a map induced by anautomorphism of its underlying graph.

Lemma 3.2.1 Let Γ be a connected graph and g ∈ AutΓ. If there is a map M ∈E L(Γ) such that the induced action g∗ ∈ AutM, then for ∀(u, v), (x, y) ∈ E (Γ),

[lg(u), lg(v)] = [lg(x), lg(y)] = constant,

where, lg(w) denotes the length of the cycle containing the vertex w in the cycledecomposition of g.

Proof According to the Lemma 2.2.1, we know that the length of a quadricelluv+ or uv− under the action g∗ is [lg(u), lg(v)]. Since g∗ is an automorphism of map,therefore, g∗ is semi-regular. Whence, we get that

[lg(u), lg(v)] = [lg(x), lg(y)] = constant.

Now we consider conditions for an induced automorphism of a map by an auto-morphism of graph to be an orientation-reversing automorphism of a map.

Lemma 3.2.2 If ξα is an automorphism of a map, then ξα = αξ.Proof Since ξα is an automorphism of a map, we know that

(ξα)α = α(ξα).

That is, ξα = αξ.

Lemma 3.2.3 If ξ is an automorphism of map M = (X α,β , P ), then ξα is semi-regular on X α,β with order o(ξ) if o(ξ) ≡ 0(mod2) and 2o(ξ) if o(ξ) ≡ 1(mod2).

Proof Since ξ is an automorphism of map by the Lemma 3.2.2, we know that thecycle decomposition of ξ can be represented by




ξ = k

(x1, x2, · · · , xk)(αx1, αx2, · · · , αxk),

where,

k denotes the product of disjoint cycles with length k = o(ξ).Therefore, if k ≡ 0(mod2), we get that

ξα =

k

(x1, αx2, x3, · · · , αxk)

and if k ≡ 1(mod2), we get that

ξα = 2k

(x1, αx2, x3,

· · ·, xk, αx1, x2, αx3,

· · ·, αxk).

Whence, ξ is semi-regular acting on X α,β . Now we can prove the following result for orientation-reversing automorphisms

of a map.

Lemma 3.2.4 For a connected graph Γ, let K be all automorphisms in AutΓ whoseextending action on X α,β , X = E (Γ), are automorphisms of maps underlying thegraph Γ. Then for ∀ξ ∈ K, o(ξ∗) ≥ 2, ξ∗α ∈ K if and only if o(ξ∗) ≡ 0(mod2).

Proof Notice that by the Lemma 3.2.3, if ξ∗ is an automorphism of map withunderlying graph Γ, then ξ∗α is semi-regular acting on X α,β .

Assume ξ∗

is an automorphism of map M = (X α,β , P ). Without loss of generality,we assume that

P = C 1C 2 · · · C k,

where,C i = (xi1, xi2, · · · , xiji) is a cycle in the decomposition of ξ|V (Γ) and xit =(ei1, ei2, · · · , eiti)(αei1, αeiti , · · · , αei2) and.

ξ|E (Γ) = (e11, e12, · · · , es1)(e21, e22, · · · , e2s2) · · · (el1, el2, · · · , elsl).

andξ∗ = C (αC −1α),

where, C = (e11, e12, · · · , es1)(e21, e22, · · · , e2s2) · · · (el1, el2, · · · , elsl). Now since ξ∗ isan automorphism of map, we get that s1 = s2 = · · · = sl = o(ξ∗) = s.

If o(ξ∗) ≡ 0(mod2), define a map M ∗ = (X α,β , P ∗), where,

P ∗ = C ∗1C ∗2 · · · C ∗k ,

where, C ∗i = (x∗i1, x∗i2, · · · , x∗iji), x∗it = (e∗i1, e∗i2, · · · , e∗iti

)(αe∗i1, αe∗iti, · · · , e∗i2) and

e∗ij = e pq. Take e∗ij = e pq if q ≡ 1(mod2) and e∗ij = αe pq if q ≡ 0(mod2). Thenwe get that M ξα = M .

Now if o(ξ∗) ≡ 1(mod2), by the Lemma 3.2.3, o(ξ∗α) = 2o(ξ∗). Therefore, anychosen quadricells (ei1, ei2,

· · ·, eiti) adjacent to the vertex xi1 for i = 1, 2,

· · ·, n,




where, n =|Γ|, the resultant map M is unstable under the action of ξα. Whence,

ξα is not an automorphism of one map with underlying graph Γ. Now we determine all automorphisms of a complete map underlying a graph K n

by applying the previous results. Recall that the automorphism group of the graphK n is just the symmetry group of degree n, that is, AutKn = S n.

Theorem 3.2.1 All orientation-preserving automorphisms of non-orientable com-plete maps of order ≥ 4 are extended actions of elements in

E [sns ]

, E [1,s

n−1s ]

,

and all orientation-reversing automorphisms of non-orientable complete maps of order ≥ 4 are extended actions of elements in

αE [(2s)

n2s ]

, αE [(2s)

42s ]

, αE [1,1,2],

where, E θ denotes the conjugatcy class containing element θ in the symmetry groupS n

Proof First, we prove that the induced permutation ξ∗ on a complete map of ordern by an element ξ ∈ S n is an cyclic order-preserving automorphism of non-orientablemap, if, and only if,

ξ ∈ E sns E [1,sn−1s ]

Assume the cycle index of ξ is [1k1, 2k2,...,nkn ]. If there exist two integers ki, k j =0, and i, j ≥ 2, i = j, then in the cycle decomposition of ξ , there are two cycles

(u1, u2,...,ui) and (v1, v2,...,v j ).

Since[lξ(u1), lξ(u2)] = i and [lξ(v1), lξ(v2)] = j

and i = j, we know that ξ∗ is not an automorphism of embedding by the Lemma2.5. Whence, the cycle index of ξ must be the form of [1k, sl].

Now if k ≥ 2, let (u), (v) be two cycles of length 1 in the cycle decomposition of ξ. By the Lemma 2.5, we know that

[lξ(u), lξ(v)] = 1.

If there is a cycle (w,...) in the cycle decomposition of ξ whose length greater orequal to two, we get that

[lξ(u), lξ(w)] = [1, lξ(w)] = lξ(w).

According to the Lemma 3.2.1, we get that lξ(w) = 1, a contradiction. Therefore,the cycle index of ξ must be the forms of [sl] or [1, sl]. Whence,sl = n or sl + 1 = n.




Calculation shows that l = n

sor l = n−1

s. That is, the cycle index of ξ is one of the

following three types [1n], [1, sn−1s ] and [s

ns ] for some integer s .

Now we only need to prove that for each element ξ in E [1,s

n−1s ]

and E [sns ]

, there

exists an non-orientable complete map M of order n with the induced permutationξ∗ being its cyclic order-preserving automorphism of surface. The discussion aredivided into two cases.

Case 1 ξ ∈ E [sns ]

Assume the cycle decomposition of ξ being ξ = (a,b, · · · , c) · · · (x,y, · · · , z) · · · (u,v,· · · , w), where, the length of each cycle is k, and 1 ≤ a,b, · · · ,c,x,y, · · · ,z ,u ,v , · · · , w ≤n . In this case, we can construct a non-orientable complete map M 1 = (

X 1

α,β ,

P 1)

as follows.

X 1α,β = i j+ : 1 ≤ i, j ≤ n, i( ji j− : 1 ≤ i, j ≤ n, i = j,

P 1 =

x∈a,b,···,c,···,x,y,···,z,u,v,···,w

(C (x))(αC (x)−1α),

where,

C (x) = (xa+,

· · ·, xx∗,

· · ·, xu+, xb+, xy+,

· · ·,

· · ·, xv+, xc+,

· · ·, xz+,

· · ·, xw+),

where xx∗ denotes an empty position and

αC (x)−1α = (xa−, xw−, · · · , xz−, · · · , xc−, xv−, · · · , xb−, xu−, · · · , xy−, · · ·).

It is clear that M ξ∗

1 = M 1. Therefore, ξ∗ is an cyclic order-preserving automor-phism of the map M 1.

Case 2 ξ ∈ E [1,s

n−1s ]

We assume the cycle decomposition of ξ being

ξ = (a,b,...,c)...(x,y,...,z)...(u,v,...,w)(t),

where, the length of each cycle is k beside the final cycle, and 1 ≤ a,b...c,x,y...,z,u,v,...,w,t ≤ n . In this case, we construct a non-orientable complete map M 2 =(X 2α,β , P 2) as follows.

X 2α,β = i j+ : 1 ≤ i, j ≤ n, i = ji j− : 1 ≤ i, j ≤ n, i = j,

P 2 = (A)(αA−1)

x∈a,b,...,c,...,x,y,...z,u,v,...,w

(C (x))(αC (x)−1α),




where,

A = (ta+, tx+,...tu+, tb+, ty+,...,tv+,...,tc+, tz+,...,tw+)

and

αA−1α = (ta−, tw−,...tz−, tc−, tv−,...,ty−,...,tb−, tu−,...,tx−)

and

C (x) = (xa+,...,xx∗,...,xu+, xb+,...,xy+,...,xv+,...,xc+,...,xz+,...,xw+)

and

αC (x)−1α = (xa−, xw−,..,xz−,...,xc−,...,xv−,...,xy−,...,xb−, xu−,...).

It is also clear that M ξ∗

2 = M 2. Therefore, ξ∗ is an automorphism of a map M 2 .Now we consider the case of orientation-reversing automorphism of a complete

map. According to the Lemma 3.2.4, we know that an element ξα, where, ξ ∈ S n,is an orientation-reversing automorphism of a complete map, only if,

ξ ∈ E [k

n1k ,(2k)

n−n12k ]

.

Our discussion is divided into two parts.

Case 3 n1 = n

Without loss of generality, we can assume the cycle decomposition of ξ has thefollowing form in this case.

ξ = (1, 2, · · · , k)(k + 1, k + 2, · · · , 2k) · · · (n − k + 1, n − k + 2, · · · , n).

Subcase 3.1 k

≡1(mod2) and k > 1

According to the Lemma 3.2.4, we know that ξ∗α is not an automorphism of map since o(ξ∗) = k ≡ 1(mod2).

Subcase 3.2 k ≡ 0(mod2)

Construct a non-orientable map M 3 = (X 3α,β , P 3), where X 3 = E (K n) and

P 3 =

i∈1,2,···,n

(C (i))(αC (i)−1α),

where, if i ≡ 1(mod2), then




C (i) = (i1+, ik+1+, · · · , in−k+1+, i2+, · · · , in−k+2+, · · · , ii∗, · · · , ik+, i2k+, · · · , in+),

αC (i)−1α = (i1−, in−, · · · , i2k−, ik−, · · · , ik+1−)

and if i ≡ 0(mod2), then

C (i) = (i1−, ik+1−, · · · , in−k+1−, i2−, · · · , in−k+2−, · · · , ii∗, · · · , ik−, i2k−, · · · , in−),

αC (i)−1α = (i1+, in+, · · · , i2k+, ik+, · · · , ik+1+).Where, ii∗ denotes the empty position, for example, (21, 22∗, 23, 24, 25) = (21, 23, 24, 25).

It is clear that P ξα3 = P 3, that is, ξα is an automorphism of map M 3.

Case 4 n1 = n

Without loss of generality, we can assume that

ξ = (1, 2, · · · , k)(k + 1, k + 2, · · · , n1) · · · (n1 − k + 1, n1 − k + 2, · · · , n1)

× (n1 + 1, n1 + 2, · · · , n1 + 2k)(n1 + 2k + 1, · · · , n1 + 4k) · · · (n − 2k + 1, · · · , n)


Consider the orbits of 12+ and n1 + 2k + 11+ under the action of < ξα >, we getthat

|orb((12+)<ξα>)| = k

and

|orb(((n1 + 2k + 1)1+)<ξα>)| = 2k.

Contradicts to the Lemma 3.2.1.


In this case, if k = 1, then k ≥ 3. Similar to the discussion of the Subcase 3.1,we know that ξα is not an automorphism of complete map. Whence, k = 1 and

ξ ∈ E [1n1 ,2n2 ].

Without loss of generality, assume that

ξ = (1)(2)· · ·

(n1)(n

1+ 1, n

1+ 2)(n

1+ 3, n

1+ 4)

· · ·(n

1+ n

2 −1, n

1+ n

2).




If n2 ≥

2, and there exists a map M = (X α,β

,P

), assume the vertex v1

in M being

v1 = (1l12+, 1l13+, · · · , 1l1n+)(1l12−, 1l1n−, · · · , 1l13−)

where, l1i ∈ +2, −2, +3, −3, · · · , +n, −n and l1i = l1 j if i = j.Then we get that

(v1)ξα = (1l12−, 1l13−, · · · , 1l1n−)(1l12+, 1l1n+, · · · , 1l13+) = v1.

Whence, ξα is not an automorphism of map M . A contradiction.Therefore, n2 = 1. Similarly, we can also get that n1 = 2. Whence, ξ =

(1)(2)(34) and n = 4. We construct a stable non-orientable map M 4 under theaction of ξα as follows.

M 4 = (X 4α,β , P 4),

where,

P 4 = (12+, 13+, 14+)(21+, 23+, 24+)(31+, 32+, 34+)(41+, 42+, 43+)

× (12−, 14−, 13−)(21−, 24−, 23−)(31−, 34−, 32−)(41−, 43−, 42−).

Therefore, all orientation-preserving automorphisms of non-orientable completemaps are extended actions of elements in

E [sns ], E

[1,sn−1s ]

and all orientation-reversing automorphisms of non-orientable complete maps areextended actions of elements in

αE [(2s)

n2s ]

, αE [(2s)

42s ]

αE [1,1,2].

This completes the proof. According to the Rotation Embedding Scheme for orientable embedding of a

graph, First presented by Heffter in 1891 and formalized by Edmonds in [17], eachorientable complete map is just the case of eliminating the sign + and - in ourrepresentation for complete map. Whence,we have the following result for an auto-morphism of orientable surfaces, which is similar to the Theorem 3.2.1.

Theorem 3.2.2 All orientation-preserving automorphisms of orientable completemaps of order ≥ 4 are extended actions of elements in

E [sns ]

, E [1,s

n−1s ]




and all orientation-reversing automorphisms of orientable complete maps of order ≥4 are extended actions of elements in

αE [(2s)

n2s ]

, αE [(2s)

42s ]

, αE [1,1,2],

where,E θ denotes the conjugatcy class containing θ in S n.

Proof The proof is similar to that of the Theorem 3.2.1. For completion, we onlyneed to construct orientable maps M Oi , i = 1, 2, 3, 4, to replace the non-orientablemaps M i, i = 1, 2, 3, 4 in the proof of the Theorem 3.2.1.

In fact, for orientation-preserving case, we only need to take M O1 , M O2 to be theresultant maps eliminating the sign + and - in M 1, M 2 constructed in the proof of

the Theorem 3.2.1.For the orientation-reversing case, we take M O3 = (E (K n)α,β , P O3 ) with

P 3 =

i∈1,2,···,n

(C (i)),

where, if i ≡ 1(mod2), then

C (i) = (i1, ik+1, · · · , in−k+1, i2, · · · , in−k+2, · · · , ii∗, · · · , ik, i2k, · · · , in),

and if i ≡ 0(mod2), then

C (i) = (i1, ik+1, · · · , in−k+1, i2, · · · , in−k+2, · · · , ii∗, · · · , ik, i2k, · · · , in)−1,

where ii∗ denotes the empty position and M O4 = (E (K 4)α,β , P 4) with

P 4 = (12, 13, 14)(21, 23, 24)(31, 34, 32)(41, 42, 43).

It can be shown that (M Oi )ξ∗α = M Oi for i = 1, 2, 3 and 4.

§3. The automorphisms of a semi-regular graph on surfaces

A graph is called a semi-regular graph if it is simple and its automorphism groupaction on its ordered pair of adjacent vertices is fixed-free, which is considered in[43], [50] for enumeration its non-equivalent embeddings on surfaces. A map under-lying a semi-regular graph is called a semi-regular map. We determine all automor-phisms of maps underlying a semi-regular graph in this section.

Comparing with the Theorem 3.1.2, we get a necessary and sufficient conditionfor an automorphism of a graph being that of a map.

Theorem 3.3.1 For a connected graph Γ, an automorphism ξ ∈ AutΓ is an orientation-preserving automorphism of a non-orientable map underlying the graph Γ iff ξ issemi-regular acting on its ordered pairs of adjacent vertices.




Proof According to the Lemma 2.2.1, if ξ∈

AutΓ is an orientation-preservingautomorphism of a map M underlying graph Γ, then ξ is semi-regular acting on itsordered pairs of adjacent vertices.

Now assume that ξ ∈ AutΓ is semi-regular acting on its ordered pairs of adjacentvertices. Denote by ξ|V (Γ), ξ|E (Γ)β

the action of ξ on V (Γ) and on its ordered pairsof adjacent vertices,respectively. By the given condition, we can assume that

ξ|V (Γ) = (a,b, · · · , c) · · · (g,h, · · · , k) · · · (x,y, · · · , z)

and

ξ

|E (Γ)β

= C 1· · ·

C i· · ·

C m,

where, sa|C (a)| = · · · = sg|C (g)| = · · · = sx|C (x)|, and C(g) denotes the cyclecontaining g in ξ|V (Γ) and

C 1 = (a1, b1, · · · , c1, a2, b2, · · · , c2, · · · , asa, bsa, · · · , csa),

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ,

C i = (g1, h1, · · · , k1, g2, k2, · · · , k2, · · · , gsg , hsg , · · · , ksg),

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ,

C m = (x1, y1, · · · , z1, · · · , x2, y2, · · · , z2, · · · , xsx, ysx, · · · , zsx).

Now for∀

ξ, ξ∈

AutΓ. We construct a stable map M = (X α,β

,P

) under theaction of ξ as follows.

X = E (Γ)

and

P =

g∈T V ξ

x∈C (g)

(C x)(αC −1x ).

Assume that u = ξf (g), and

N Γ(g) = gz1

, gz2

, · · · , gzl

.Obviously, all degrees of vertices in C (g) are same. Notices that ξ|N Γ(g) is circular

acting on N Γ(g) by the Theorem 3.1.2. Whence, it is semi-regular acting on N Γ(g).Without loss of generality, we can assume that

ξ|N Γ(g) = (gz1, gz2, · · · , gzs)(gzs+1, gzs+2, · · · , gz2s) · · · (gz(k−1)s+1, gz(k−1)s+2, · · · , gzks),

where, l = ks. Choose

C g

= (gz1+, gzs+1+,· · ·

, gz(k−1)s+1+, gz2+, gzs+2+,· · ·

, gzs+, gz2s,· · ·

, gzks+).




Then,

C x = (xz1+, xzs+1+, · · · , xz(k−1)s+1+, xz2+, xzs+2+, · · · , xzs+, xz2s, · · · , xzks+),

where,

xzi+ = ξf (gzi+),

for i = 1, 2, · · · ,ks. and

αC −1x = (αxz1+, αxzs+1+,

· · ·, αxz(k−1)s+1+, αxzs+, αxz2s,

· · ·, αxzks+).

Immediately, we get that M ξ = ξM ξ−1 = M by this construction. Whence, ξ isan orientation-preserving automorphism of the map M .

By the Rotation Embedding Scheme, eliminating α on each quadricell in Tutte’srepresentation of an embedding induces an orientable embedding with the sameunderlying graph. Since an automorphism of an embedding is commutative with αand β , we get the following result for the orientable-preserving automorphisms of the orientable maps underlying a semi-regular graph.

Theorem 3.3.2 If Γ is a connected semi-regular graph, then for ∀ξ ∈ AutΓ, ξ is an orientation-preserving automorphism of orientable maps underlying the graph Γ.

According to the Theorem 3.3.1 and 3.3.2, if Γ is semi-regular, i.e., each auto-morphism acting on the ordered pairs of adjacent vertices in Γ is fixed-free, thenevery automorphism of the graph Γ is an orientation-preserving automorphism of orientable maps and non-orientable maps underlying the graph Γ. We restated thisresult as the following.

Theorem 3.3.3 If Γ is a connected semi-regular graph, then for ∀ξ ∈ AutΓ, ξis an orientation-preserving automorphism of orientable maps and non-orientablemaps underlying the graph Γ.

Notice that if ς ∗ is an orientation-reversing automorphisms of a map, then ς ∗α

is an orientation-preserving automorphism of the same map. By the Lemma 3.2.4,if τ is an automorphism of maps underlying a graph Γ, then τ α is an automorphismof maps underlying this graph if and only if τ ≡ 0(mod2). Whence, we have thefollowing result for the automorphisms of maps underlying a semi-regular graph

Theorem 3.3.4 Let Γ be a semi-regular graph. Then all the automorphisms of orientable maps underlying the graph Γ are

g|X α,β and αh|X α,β , g , h ∈ AutΓ with h ≡ 0(mod2).

and all the automorphisms of non-orientable maps underlying the graph Γ are also





Theorem 3.3.4 will be used in the Chapter 4 for the enumeration of unrootedmaps on surfaces underlying a semi-regular graph.

§4. The automorphisms of one face maps

There is a well-know result for the automorphisms of a map and its dual in

topological graph theory, i.e., the automorphism group of a map is same as its dual.Therefore, for determining the automorphisms of one face maps, we can determinethem by the automorphisms of a bouquet Bn on surfaces.

A map underlying a graph Bn, n ≥ 1 has the form Bn = (X α,β , P n) with X =E (Bn) = e1, e2, · · · , en and

P n = (x1, x2, · · · , x2n)(αx1, αx2n, · · · , x2)

where, xi ∈ X,βX or αβX and satisfying the condition (Ci) and (Cii) in the Section2.2 of Chapter 1. For a given bouquet Bn with n edges, its semi-arc automorphismgroup is

Aut12

Bn = S n[S 2].

Form group theory, we know that each element in S n[S 2] can be represented by(g; h1, h2, · · · , hn) with g ∈ S n and hi ∈ S 2 = 1, αβ for i = 1, 2, · · · , n. The actionof (g; h1, h2, · · · , hn) on a map Bn underlying a graph Bn by the following rule:

if x ∈ ei, αei, βei,αβei, then (g; h1, h2, · · · , hn)(x) = g(hi(x)).

For example, if h1 = αβ , then,(g; h1, h2, · · · , hn)(e1) = αβg(e1), (g; h1, h2, · · · , hn)(αe1)= βg(e1), (g; h1, h2, · · · , hn)(βe1) = αg(e1) and (g; h1, h2, · · · , hn)(αβe1) = g(e1).

The following result for automorphisms of a map underlying a graph Bn is obvi-ous.

Lemma 3.4.1 Let (g; h1, h2, · · · , hn) be an automorphism of a map Bn underlying a graph Bn. Then

(g; h1, h2, · · · , hn) = (x1, x2,...,x2n)k,

and if (g; h1, h2, · · · , hn)α is an automorphism of a map Bn, then

(g; h1, h2, · · · , hn)α = (x1, x2, · · · , x2n)k

for some integer k, 1 ≤ k ≤ n. Where, xi ∈ e1, e2, · · · , en, i = 1, 2, · · · , 2n and x

i = x

jif i

= j.




Analyzing the structure of elements in the group S n

[S 2], we get the automor-

phisms of maps underlying a graph Bn by the Theorem 3.3.1 and 3.3.2 as follows.

Theorem 3.4.1 Let Bn be a bouquet with n edges 1, 2, · · · , n. Then the automor-phisms (g; h1, h2, · · · , hn) of orientable maps underlying a Bn, n ≥ 1, are respective

(O1) g ∈ E [k

nk ]

, hi = 1, i = 1, 2, · · · , n;

(O2) g ∈ E [k

nk ]

and if g =n/ki=1

(i1, i2, · · · ik), where i j ∈ 1, 2, · · · , n,n/k ≡0(mod2), then hi1 = (1, αβ ), i = 1, 2, · · · , n

k and hij = 1 for j ≥ 2;

(O3) g ∈ E [k2s,(2k)

n−2ks2k ]

and if g =2s

i=1(i1, i2, · · · ik)

(n−2ks)/2k

j=1(e j1, e j2, · · · , e j2k),

where i j, e jt ∈ 1, 2, · · · , n, then hi1 = (1, αβ ), i = 1, 2, · · · , s , hil = 1 for l ≥2 and h jt = 1 for t = 1, 2, · · · , 2kand the automorphisms (g; h1, h2, · · · , hn) of non-orientable maps underlying a Bn, n≥ 1, are respective

(N 1) g ∈ E [k

nk ]

, hi = 1, i = 1, 2, · · · , n;

(N 2) g ∈ E [k

nk ]

and if g =n/ki=1

(i1, i2, · · · ik), where i j ∈ 1, 2, · · · , n,n/k ≡0(mod2), then hi1 = (1, αβ ), (1, β ) with at least one hi01(1, β ), i = 1, 2, · · · , n

kand hij =

1 for j ≥ 2;

(N 3) g

∈ E [k2s,(2k)

n−2ks2k ]

and if g =2s

i=1

(i1, i2,

· · ·ik)

(n−2ks)/2k

j=1

(e j1, e j2,

· · ·, e j2k),

where i j , e jt ∈ 1, 2, · · · , n, then hi1 = (1, αβ ), (1, β ) with at least one hi01 =(1, β ), i = 1, 2, · · · , s , hil = 1 for l ≥ 2 and h jt = 1 for t = 1, 2, · · · , 2k,where, E θ denotes the conjugacy class in the symmetry group S n containing theelement θ.

Proof By the structure of the group S n[S 2], it is clear that the elements in thecases (i), (ii) and (iii) are its all semi-regular elements. We only need to constructan orientable or non-orientable map Bn = (X α,β , P n) underlying Bn stable under theaction of an element in each case.

(i) g =n/k

i=1

(i1, i2,· · ·

ik) and hi = 1, i = 1, 2,· · ·

, n, where i j∈

1, 2,· · ·

, n

.

Choose

X 1α,β =n/ki=1

K i1, i2, · · · , ik,

where K = 1,α,β ,αβ and

P 1n = C 1(αC −11 α−1)

with




C 1 = ( 11, 21, · · · , (n

k)1, αβ 11, αβ 21, · · · , αβ (

n

k)1, 12, 22, · · · , (

n

k)2,

αβ 12, αβ 22, · · · , αβ (n

k)2, · · · , 1k, 2k, · · · , (

n

k)k, αβ 1k, αβ 1k, · · · , αβ (

n

k)k).

Then the map B1n = (X 1α,β , P 1n) is an orientable map underlying the graph Bn and

stable under the action of (g; h1, h2, · · · , hn).For the non-orientable case, we can chose

C 1 = ( 11, 21,

· · ·, (

n

k

)1, β 11, β 21,

· · ·, β (

n

k

)1, 12, 22,

· · ·, (

n

k

)2,

β 12, β 22, · · · , β (n

k)2, · · · , 1k, 2k, · · · , (

n

k)k, β 1k, β 1k, · · · , β (

n

k)k).

Then the map B1n = (X 1α,β , P 1n) is a non-orientable map underlying the graph Bn

and stable under the action of (g; h1, h2, · · · , hn).

(ii) g =n/ki=1

(i1, i2, · · · ik), hi = (1, β ) or (1, αβ ), i = 1, 2, · · · , n, nk

≡ 0(mod2),

where i j ∈ 1, 2, · · · , n.

If hi1 = (1, αβ ) for i = 1, 2, · · · , nk

and hit = 1 for t ≥ 2. Then

(g; h1, h2, · · · , hn) =n/ki=1

(i1,αβi2, · · · αβik,αβi1, i2, · · · , ik).

Similar to the case (i), we take X 2α,β = X 1α,β and

P 2n = C 2(αC −12 α−1)

with

C 2 = ( 11, 21, · · · , (n

k)1, αβ 12, αβ 22, · · · , αβ (

n

k)2, αβ 1k, αβ 2k,

· · · , αβ (n

k )k, αβ 11, αβ 21, · · · , αβ (n

k )1, 12, 22, · · · , (n

k )2, · · · , 1k, 2k, · · · , (n

k )k).


stable under the action of (g; h1, h2, · · · , hn). For the non-orientable case, the con-struction is similar. It only need to replace each element αβi j by βi j in the con-struction of the orientable case if hij = (1, β ).

(iii) g =2s

i=1(i1, i2, · · · ik)

(n−2ks)/2k j=1

(e j1, e j2, · · · , e j2k) and hi1 = (1, αβ ), i = 1, 2, · · · , s , hil

1 for l ≥ 2 and h jt = 1 for t = 1, 2, · · · , 2k

In this case, we know that




(g; h1, h2, · · · , hn) =s

i=1

(i1,αβi2, · · · αβik,αβi1, i2, · · · , ik)(n−2ks)/2k

j=1

(e j1, e j2, · · · , e j2k).

Denote by p the number (n − 2ks)/2k. We construct an orientable map B3n =

(X 3α,β , P 3n) underlying Bn stable under the action of (g; h1, h2, · · · , hn) as follows.Take X 3α,β = X 1α,β and

P 3n = C 3(αC −13 α−1)

with

C 3 = ( 11 , 21, · · · , s1, e11 , e21 , · · · , e p1, αβ 12, αβ 22, · · · ,αβs2,

e12, e22, · · · , e p2, · · · , αβ 1k, αβ 2k, · · · ,αβsk, e1k , e2k , · · · ,

e pk , αβ 11, αβ 21, · · · ,αβs1, e1k+1, e2k+1

, · · · , e pk+1, 12, 22, · · · ,

s2, e1k+2, e2k+2

, · · · , e pk+2, · · · , 1k, 2k, · · · , sk, e12k , e22k , · · · , e p2k).


stable under the action of (g; h1, h2, · · · , hn).Similarly, replacing each element αβi j by βi j in the construction of the orientable

case if hij = (1, β ), a non-orientable map underlying the graph Bn and stable underthe action of (g; h1, h2, · · · , hn) can be also constructed. This completes the proof.

We use the Lemma 3.4.1 for the enumeration of unrooted one face maps onsurfaces in the next chapter.



Chapter4

Application to the Enumeration of Unrooted Mapsand s-manifolds

All the results gotten in the Chapter 3 is more useful for the enumeration of unrooted maps on surfaces underlying a graph. According to the theory in Chapter1, the enumeration of unrooted maps on surfaces underlying a graph can be carriedout by the following programming:

STEP 1. Determine all automorphisms of maps underlying a graph;

STEP 2. Calculate the the fixing set F ix(ς ) for each automorphism ς of a map;

STEP 3. Enumerate the unrooted maps on surfaces underlying a graph byCorollary 1.3.1.

The advantage of this approach is its independence of the orientability, which enablesus to enumerate orientable or non-orientable maps on surafces. We present theenumeration results by this programming in this chapter.

§1. The enumeration of unrooted complete maps on surfaces

We first consider a permutation and its stabilizer . A permutation with thefollowing form (x1, x2,...,xn)(αxn, αx2,...,αx1) is called a permutation pair . Thefollowing result is obvious.

Lemma 4.1.1 Let g be a permutation on the set Ω = x1, x2,...,xn such that gα = αg. If

g(x1, x2,...,xn)(αxn, αxn−1,...,αx1)g−1 = (x1, x2,...,xn)(αxn, αxn−1,...,αx1),

then

g = (x1, x2,...,xn)k

and if

gα(x1, x2,...,xn)(αxn, αxn−1,...,αx1)(gα)−1 = (x1, x2,...,xn)(αxn, αxn−1,...,αx1),

then

gα = (αxn, αxn−1,...,αx1)k

for some integer k, 1≤

k≤

n.



Chapter 4 Application to the Enumeration of Unrooted Maps and s-Manifolds 69

Lemma 4.1.2 For each permutation g, g∈ E [k

n

k ]satisfying gα = αg on the set

Ω = x1, x2,...,xn, the number of stable permutation pairs in Ω under the action of g or gα is

2φ(k)(n − 1)!

|E [k

nk ]

| ,

where φ(k) denotes the Euler function.

Proof Denote the number of stable pair permutations under the action of g orgα by n(g) and C the set of pair permutations. Define the set A = (g, C )|g ∈

E [k

nk ]

, C

∈ Cand C g = C or C gα = C

. Clearly, for

∀g1, g2

∈ E [k

nk ]

, we have

n(g1) = n(g2). Whence, we get that

|A| = |E [k

nk ]

|n(g). (4.1.1)

On the other hand, by the Lemma 4.1.1, for any permutation pair C = (x1, x2,...,xn)(αxn, αxn−1,...,αx1), since C is stable under the action of g, there must be g =(x1, x2,...,xn)l or gα = (αxn, αxn−1,...,αx1)l, where l = s n

k , 1 ≤ s ≤ k and (s, k) =1. Therefore, there are 2φ(k) permutations in E

[knk ]

acting on it stable. Whence, we

also have

|A

|= 2φ(k)

|C|. (4.1.2)

Combining (4.1.1) with (4.1.2), we get that

n(g) =2φ(k)|C||E

[knk ]

| =2φ(k)(n − 1)!

|E [k

nk ]

| .

Now we can enumerate the unrooted complete maps on surfaces.

Theorem 4.1.1 The number nL(K n) of complete maps of order n ≥ 5 on surfacesis

nL(K n) = 12

(k|n

+ k|n,k≡0(mod2)

) 2α(n,k)(n − 2)!n

k

knk ( n

k )!+

k|(n−1),k=1

φ(k)2β (n,k)(n − 2)!n−1

k

n − 1,

where,

α(n, k) =

n(n−3)

2k , if k ≡ 1(mod2);n(n−2)

2k, if k ≡ 0(mod2),

and

β (n, k) = (n−1)(n−2)

2k, if k ≡ 1(mod2);

(n−1)(n−3)

2k , if k ≡ 0(mod2).




and nL(K 4) = 11.

Proof According to (1.3.3) in the Corollary 1.3.1 and the Theorem 3.2.1 for n ≥ 5,we know that

nL(K n) =1

2|AutKn| × (

g1∈E [knk ]

|Φ(g1)| +

g2∈E [(2s)

n2s ]

|Φ(g2α)|

+

h∈E [1,k

n−1k ]

|Φ(h)|)

=1

2n! × (k|n

|E [knk ]||Φ(g1)| + l|n,l≡0(mod2)

|E [lnl ]||Φ(g2α)|+

l|(n−1)

|E [1,l

n−1l ]

||Φ(h)|),

where, g1 ∈ E [k

nk ]

, g2 ∈ E [lnl ]

and h ∈ E [1,k

n−1k ]

are three chosen elements.

Without loss of generality, we assume that an element g, g ∈ E [k

nk ]

has the fol-

lowing cycle decomposition.

g = (1, 2,...,k)(k + 1, k + 2,..., 2k)...((n

k −1)k + 1, (

n

k −1)k + 2,...,n)

and

P =

1×

2,

where,

1

= (1i21, 1i31,..., 1in1)(2i12, 2i32, ..., 2in2)...(ni1n, ni2n ,...,ni(n−1)n),

and

2= α(1

−1)α−1,

being a complete map which is stable under the action of g, where sij ∈ k+, k−|k =1, 2,...n.

Notice that the quadricells adjacent to the vertex ”1” can make 2n−2(n−2)!differentpair permutations and for each chosen pair permutation, the pair permutations ad-

jacent to the vertices 2, 3,...,k are uniquely determined since P is stable under theaction of g.

Similarly, for each given pair permutation adjacent to the vertex k + 1, 2k +1, ..., ( n

k− 1)k + 1, the pair permutations adjacent to k + 2, k + 3, ..., 2k and 2k +

2, 2k +3,..., 3k and,...,and ( nk−1)k +2, ( n

k−1)k +3,...n are also uniquely determined

becauseP

is stable under the action of g.




Now for an orientable embedding M 1

of K n

, all the induced embeddings byexchanging two sides of some edges and retaining the others unchanged in M 1 arethe same as M 1 by the definition of maps. Whence, the number of different stableembeddings under the action of g gotten by exchanging x and αx in M 1 for x ∈U, U ⊂ X β , where X β =

x∈E (K n)

x,βx , is 2g(ε)−nk , where g(ε) is the number of

orbits of E (K n) under the action of g and we substract nk because we can chosen

12+, k + 11+, 2k + 11+, · · · , n − k + 11+ first in our enumeration.Notice that the length of each orbit under the action of g is k for ∀x ∈ E (K n) if

k is odd and is k2 for x = ii+ k

2 , i = 1, k + 1, · · · , n − k + 1, or k for all other edges if k is even. Therefore, we get that

g(ε) = ε(K n)

k , if k ≡ 1(mod2);ε(K n)−

n2

k , if k ≡ 0(mod2).

Whence, we have that

α(n, k) = g(ε) − n

k=

n(n−3)

2k , if k ≡ 1(mod2);n(n−2)

2k, if k ≡ 0(mod2),

and

|Φ(g)

|= 2α(n,k)(n

−2)!

nk , (4.1.3)

Similarly, if k ≡ 0(mod2), we get also that

|Φ(gα)| = 2α(n,k)(n − 2)!nk (4.1.4)

for an chosen element g,g ∈ E [k

nk ]

.

Now for ∀h ∈ E [1,k

n−1k ]

, without loss of generality, we assume that h = (1, 2,...,k)

(k + 1, k + 2,..., 2k)...(( n−1k

− 1)k + 1, ( n−1k

− 1)k + 2,..., (n − 1))(n). Then the abovestatement is also true for the complete graph K n−1 with the vertices 1, 2, · · · , n − 1.Notice that the quadricells n1+, n2+, · · · , nn−1+ can be chosen first in our enumera-tion and they are not belong to the graph K n−1. According to the Lemma 4.1.2, weget that

|Φ(h)| = 2β (n,k)(n − 2)!n−1k × 2φ(k)(n − 2)!

|E [1,k

n−1k ]

| , (4.1.5)

Where

β (n, k) = h(ε) =

ε(K n−1)

k− n−1

k = (n−1)(n−4)2k , if k ≡ 1(mod2);

ε(K n−1)k

− n−1k

= (n−1)(n−3)2k

, if k ≡ 0(mod2).

Combining (4.1.3) − (4.1.5), we get that




nL(K n) =1

2n!× (

k|n

|E [k

nk ]

||Φ(g0)| +

l|n,l≡0(mod2)

|E [lnl ]||Φ(g1α)|

+

l|(n−1)

|E [1,l

n−1l ]

||Φ(h)|)

=1

2n!× (

k|n

n!2α(n,k)(n − 2)!nk

knk ( n

k)!

+

k|n,k≡0(mod2)

n!2α(n,k)(n − 2)!nk

knk ( n

k)!

+

k|(n−1),k=1

n!

kn−1k ( n−1

k )!× 2φ(k)(n − 2)!2β (n,k)(n − 2)!

n−1k

(n−1)!

k

n−1k

(

n−1

k )!

)

=1

2(k|n

+

k|n,k≡0(mod2)

)2α(n,k)(n − 2)!

nk

knk ( n

k )!+

k|(n−1),k=1

φ(k)2β (n,k)(n − 2)!n−1k

n − 1.

For n = 4, similar calculation shows that nL(K 4) = 11 by consider the fixing setof permutations in E

[s4s ]

,E [1,s

3s ]

, E [(2s)

42s ]

,αE [(2s)

42s ]

and αE [1,1,2].

For the orientable case, we get the number nO(K n) of orientable complete mapsof order n as follows.

Theorem 4.1.2 The number nO((K n) of complete maps of order n ≥ 5 on orientable

surfaces is

nO(K n) =1

2(k|n

+

k|n,k≡0(mod2)

)(n − 2)!

nk

knk ( n

k )!+

k|(n−1),k=1

φ(k)(n − 2)!n−1k

n − 1.

and n(K 4) = 3.

Proof According to Tutte’s algebraic representation of map, a map M = (X α,β , P )is orientable if and only if for ∀x ∈ X α,β , x and αβx are in a same orbit of X α,β underthe action of the group ΨI =< αβ, P >. Now applying (1.3.1) in the Corollary 1.3.1

and the Theorem 3.2.1, Similar to the proof of the Theorem 4.1.1, we get the numbernO(K n) for n ≥ 5 as follows

nO(K n) =1

2(k|n

+

k|n,k≡0(mod2)

)(n − 2)!

nk

knk ( n

k)!

+

k|(n−1),k=1

φ(k)(n − 2)!n−1k

n − 1.

and for the complete graph K 4, calculation shows that n(K 4) = 3. Notice that nO(K n) + nN (K n) = nL(K n). Therefore, we can also get the number

nN (K n) of unrooted complete maps of order n on non-orientable surfaces by theTheorem 4.1.1 and the Theorem 4.1.2 as follows.




Theorem 4.1.3 The number nN (K n

) of unrooted complete maps of order n, n≥

5on non-orientable surfaces is

nN (K n) =1

2(k|n

+

k|n,k≡0(mod2)

)(2α(n,k) − 1)(n − 2)!

nk

knk ( n

k)!

+

k|(n−1),k=1

φ(k)(2β (n,k) − 1)(n − 2)!n−1k

n − 1,

and nN (K 4) = 8. Where, α(n, k) and β (n, k) are same as in Theorem 4.1.1.

Fig.4.1¸For n = 5, calculation shows that nL(K 5) = 1080 and nO(K 5) = 45 based on

the Theorem 4.1.1 and 4.1.2. For n = 4, there are 3 unrooted orientable maps and8 non-orientable maps shown in the Fig.4.1. All the 11 maps of K 4 on surfaces arenon-isomorphic.




Now consider the action of orientation-preserving automorphisms of completemaps, determined in the Theorem 3.2.1 on all orientable embeddings of a completegraph of order n. Similar to the proof of the Theorem 4.1.2, we can get the numberof non-equivalent embeddings of a complete graph of order n, which has been donein [43] and the result gotten is same as the result of Mull et al in [54].

§2. The enumeration of a semi-regular graph on surfaces

The non-equivalent embeddings of a semi-regular graph on surfaces are enumer-ated in the reference [50]. Here, we enumerate the unrooted maps underlying asemi-regular graph on orientable or non-orientable surfaces.

The following lemma is implied in the proof of the Theorem 4.1 in [50].

Lemma 4.2.1 Let Γ = (V, E ) be a semi-regular graph. Then for ξ ∈ AutΓ

|ΦO(ξ)| =

x∈T V ξ

(d(x)

o(ξ|N Γ(x))− 1)!

and

|ΦL(ξ)

|= 2|T E

ξ|−|T V

ξ| x∈T V

ξ

(d(x)

o(ξ|N Γ(x)) −1)!,

where, T V ξ , T E

ξ are the representations of orbits of ξ acting on v(Γ) and E (Γ) ,re-spectively and ξN Γ(x) the restriction of ξ to N Γ(x).

According to the Corollary 1.3.1, we get the following enumeration results.

Theorem 4.2.1 Let Γ be a semi-regular graph. Then the numbers of unrooted mapson orientable and non-orientable surfaces underlying the graph Γ are

nO(Γ) =1

|AutΓ|( ξ∈AutΓ

λ(ξ) x∈T V ξ

(d(x)

o(ξ|N Γ(x)) −1)!

and

nN (Γ) =1

|AutΓ| × ξ∈AutΓ

(2|T Eξ|−|T V

ξ| − 1)λ(ξ)

x∈T V

ξ

(d(x)

o(ξ|N Γ(x))− 1)!,

where λ(ξ) = 1 if o(ξ) ≡ 0(mod2) and 12

, otherwise.

Proof By the Corollary 1.3.1, we know that




nO(Γ) = 12|Aut 1

2Γ|

g∈Aut1

2Γ

|ΦO1 (g)|

and

nL(Γ) =1

2|Aut 12

Γ|

g∈Aut12Γ

|ΦT 1 (g)|.

According to the Theorem 3.3.4, all the automorphisms of orientable maps un-derlying the graph Γ are

g|X α,β

and αh|X α,β

, g , h ∈ AutΓ with h ≡ 0(mod2).and all the automorphisms of non-orientable maps underlying the graph Γ are also


Whence, we get the number of unrooted orientable maps by the Lemma 4.2.1 asfollows.

nO(Γ) =1

2|AutΓ|

g∈AutΓ

|ΦO1 (g)|

=1

2|AutΓ|( ξ∈AutΓ

x∈T V

ξ

(d(x)

o(ξ|N Γ(x)) − 1)!

+

ς ∈AutΓ,o(ς )≡0(mod2)

x∈T V ς

(d(x)

o(ς |N Γ(x))− 1)!)

=1

|AutΓ|(

ξ∈AutΓ

λ(ξ)

x∈T V ξ

(d(x)

o(ξ|N Γ(x))− 1)!.

Similarly, we enumerate the unrooted maps on locally orientable surface under-lying the graph Γ by (1.3.3) in the Corollary 1.3.1 as follows.

nL(Γ) =1

2|AutΓ|

g∈AutΓ

|ΦT 1 (g)|

=1

2|AutΓ|(

|T Eξ|−|T V

ξ|

ξ∈AutΓ

x∈T V

ξ

(d(x)

o(ξ|N Γ(x))− 1)!

+

ς ∈AutΓ,o(ς )≡0(mod2)

2|T Eς |−|T V ς |

x∈T V ς

(d(x)

o(ς |N Γ(x))− 1)!)

=1

|AutΓ

| ξ∈AutΓ

λ(ξ)2|T Eξ|−|T V

ξ|

x∈T V ξ

(d(x)

o(ξ

|N Γ(x))

− 1)!.




Notice that nO(Γ) + nN (Γ) = nL(Γ), we get the number of unrooted maps onnon-orientable surfaces underlying the graph Γ as follows.

nN (Γ) = nL(Γ) − nO(Γ)

=1

|AutΓ| × ξ∈AutΓ

(2|T Eξ|−|T V

ξ| − 1)λ(ξ)

x∈T V

ξ

(d(x)

o(ξ|N Γ(x))− 1)!

This completes the proof. If Γ is also a k-regular graph, we can get a simple result for the numbers of

unrooted maps on orientable or non-orientable surfaces.

Corollary 4.2.1 Let Γ be a k-regular semi-regular graph. Then the numbers of unrooted maps on orientable or non-orientable surfaces underlying the graph Γ arerespective

nO(Γ) =1

|AutΓ| × g∈AutΓ

λ(g)(k − 1)!|T V g |

and

nN (Γ) =1

|AutΓ

|

×g∈AutΓ

λ(g)(2|T Eg |−|T V g | − 1)(k − 1)!|T V g |,

where, λ(ξ) = 1 if o(ξ) ≡ 0(mod2) and 12

, otherwise.

Proof In the proof of the Theorem 4.2 in [50], it has been proved that

∀x ∈ V (Γ), o(ξN Γ(x)) = 1.

Whence, we get nO(Γ) and nN (Γ) by the Theorem 4.2.1. Similarly, if Γ = Cay(Z p : S ) for a prime p, we can also get closed formulas for

the number of unrooted maps underlying the graph Γ.

Corollary 4.2.2 Let Γ = Cay(Z p : S ) be connected graph of prime order p with

( p − 1, |S |) = 2. Then

nO(Γ, M) =(|S | − 1)! p + 2 p(|S | − 1)!

p+12 + ( p − 1)(|S | − 1)!

4 p

and

nN (Γ, M) =(2

p|S|2− p − 1)(|S | − 1)! p + 2(2

p|S|−2p−2)4 − 1) p(|S | − 1)!

p+12

2 p

+(2

|S|−22 − 1)( p − 1)(|S | − 1)!

4 p

.




Proof By the proof of the Corollary 4.1 in [50], we have known that

|T V g | =

1, if o(g) = p

p+12

, if o(g) = 2 p, if o(g) = 1

and

|T E g | =

|S |2 , if o(g) = p

p|S |4 , if o(g) = 2

p|S |2

, if o(g) = 1

Notice that AutΓ = D p (see [3][12]) and there are p elements order 2, one order1 and p − 1 order p. Whence, we have

nO(Γ, M) =(|S | − 1)! p + 2 p(|S | − 1)!

p+12 + ( p − 1)(|S | − 1)!

4 p

and

nN (Γ, M) =(2

p|S|2− p − 1)(|S | − 1)! p + 2(2

p|S|−2p−2)4 − 1) p(|S | − 1)!

p+12

2 p

+ (2|S|−2

2 − 1)( p − 1)(|S | − 1)!4 p

.

By the Corollary 4.2.1.

§3. The enumeration of a bouquet on surfaces

For any integer k, k|2n, let J k be the conjugacy class in S n[S 2] with each cycle inthe decomposition of a permutation in J k being k-cycle. According to the Corollary

1.3.1, we need to determine the numbers |ΦO

(ξ)| and |ΦL

(ξ)| for each automorphismof a map underlying a graph Bn.

Lemma 4.3.1 Let ξ =2n/ki=1

(C (i))(αC (i)α−1) ∈ J k, where, C (i) = (xi1, xi2, · · · , xik)

is a k-cycle, be a cycle decomposition. Then (i) If k = 2n, then

|ΦO(ξ)| = k2nk (

2n

k− 1)!

and if k=2n, then |ΦO(ξ)| = φ(2n).(ii) If k

≥3 and k

= 2n, then




|ΦL(ξ)| = (2k)2nk−1( 2n

k− 1)!,

and

|ΦL(ξ)| = 2n(2n − 1)!

if ξ = (x1)(x2) · · · (xn)(αx1)(αx2) · · · (αxn)(βx1)(βx2) · · · (βxn)(αβx1)(αβx2) · · · (αβxn),and

|ΦL(ξ)| = 1

if ξ = (x1,αβx1)(x2,αβx2) · · · (xn,αβxn)(αx1, βx1)(αx2, βx2) · · · (αxn, βxn), and

|ΦL(ξ)| =n!

(n − 2s)!s!

if ξ = ζ ; ε1, ε2, · · · , εn and ζ ∈ E [1n−2s,2s] for some integer s, εi = (1, αβ ) for 1 ≤ i ≤ sand ε j = 1 for s + 1 ≤ j ≤ n,where, E [1n−2s,2s] denotes the conjugate class with thetype [1n−2s, 2s] in the symmetry group S n, and

|ΦL(ξ)| = φ(2n)

if ξ = θ; ε1, ε2, · · · , εn and θ ∈ E [n1] and εi = 1 for 1 ≤ i ≤ n − 1, εn = (1, αβ ),

where, φ(t) is the Euler function.Proof (i) Notice that for a given representation of C (i), i = 1, 2, · · · , 2n

k, since

< P n,αβ > is not transitive on X α,β , there is one and only one stable orientablemap Bn = (X α,β , P n) with X = E (Bn) and P n = C (αC −1α−), where,

C = (x11, x21, · · · , x2nk1, x21, x22, · · · , x2n

k2, x1k, x2k, · · · , x 2n

kk).

Counting ways of each possible order for C (i), i = 1, 2, · · · , 2nk , and different repre-

sentations for C (i), we know that

|ΦO(ξ)

|= k

2nk (

2n

k −1)!

for k = 2n.Now if k = 2n, then the permutation itself is a map with the underlying graph

Bn. Whence, it is also an automorphism of the map with the permutation is itspower. Therefore, we get that

|ΦO(ξ)| = φ(2n)

.(ii) For k ≥ 3 and k = 2n, since the group < P n,αβ > is transitive on X α,β

or not, we can interchange C (i) by αC (i)−1α−1 for each cycle not containing the




quadricell x11

. Notice that we only get the same map if the two sides of some edgesare interchanged altogether or not. Whence, we get that

|ΦL(ξ)| = 22nk−1k

2nk−1(

2n

k− 1)! = (2k)

2nk−1(

2n

k− 1)!.

Now if ξ = (x1,αβx1)(x2,αβx2) · · · (xn,αβxn)(αx1, βx1)(αx2, βx2) · · · (αxn, βxn),there is one and only one stable map (X α,β , P 1n) under the action of ξ, where

P 1n = (x1, x2, · · · , xn,αβx1,αβx2, · · · ,αβxn)(αx1, βxn, · · · , βx1, αxn, · · · , αx1).

Which is an orientable map. Whence,

|ΦL(ξ)

|=

|ΦO(ξ)

|= 1.

If ξ = (x1)(x2) · · · (xn)(αx1)(αx2) · · · (αxn)(βx1)(βx2) · · · (βxn)(αβx1)(αβx2)· · · (αβxn), we can interchange (αβxi) with (βxi) and obtain different embeddingsof Bn on surfaces. Whence, we know that

|ΦL(ξ)| = 2n(2n − 1)!.

Now if ξ = (ζ ; ε1, ε2, · · · , εn) and ζ ∈ E [1n−2s,2s] for some integer s, εi = (1, αβ )for 1 ≤ i ≤ s and ε j = 1 for s + 1 ≤ j ≤ n, we can not interchange (xi,αβxi) with(αxi, βxi) to get different embeddings of Bn for it just interchanging the two sidesof one edge. Whence, we get that

|ΦL(ξ)| = n!1n−2s(n − 2s)!2ss!

× 2s = n!(n − 2s)!s!

.

For ξ = (θ; ε1, ε2, · · · , εn), θ ∈ E [n1] and εi = 1 for 1 ≤ i ≤ n − 1, εn = (1, αβ ), wecan not get different embeddings of Bn by interchanging the two conjugate cycles,whence, we get that

|ΦL(ξ)| = |ΦO(ξ)| = φ(2n).

This completes the proof. Recall that the cycle index of a group G, denoted by Z (G; s1, s2, · · · , sn), is

defined by ([30])

Z (G; s1, s2, · · · , sn) =1

|G|g∈G

sλ1(g)1 s

λ2(g)2 · · · sλn(g)

n ,

where, λi(g) is the number of i-cycles in the cycle decomposition of g. For thesymmetric group S n, its cycle index is known as follows:

Z (S n; s1, s2, · · · , sn) =

λ1+2λ2+···+kλk=n

sλ11 sλ2

2 · · · sλkk

1λ1λ1!2λ2λ2! · · · kλkλk!.

For example, we have that Z (S 2) =s21+s2

2 . By a result of Polya ([56]), we know thecycle index of S

n[S

2] as follows:




Z (S n[S 2]; s1, s2, · · · , s2n) =1

2nn!

λ1+2λ2+···+kλk=n

(s21+s2

2 )λ1(s22+s4

2 )λ2 · · · (s2k+s2k2 )λk

1λ1λ1!2λ2λ2! · · · kλkλk!

Now we can count maps on surfaces with an underlying graph Bn.

Theorem 4.3.1 The number nO(Bn) of non-isomorphic maps on orientable surfacesunderlying a graph Bn is

nO(Bn) =

k|2n,k=2n

k2nk−1(

2n

k− 1)!

1

(2n

k)!

∂ 2nk (Z (S n[S 2]))

∂s

2nk

k

|sk=0

+ φ(2n)∂ (Z (S n[S 2]))

∂s2n|s2n=0

Proof According to the formula (1.3.1) in the Corollary 1.3.1, we know that

nO(Bn) =1

2 × 2nn!

ξ∈S n[S 2]×≺α≻

|ΦT (ξ)|.

Now since for ∀ξ1, ξ2 ∈ S n[S 2], if there exists an element θ ∈ S n[S 2], such thatξ2 = θξ1θ−1, then we have that |ΦO(ξ1)| = |ΦO(ξ2)| and |ΦO(ξ)| = |ΦO(ξα)|. Noticethat

|ΦO(ξ)

|has been gotten by the Lemma 4.3.1. Using the Lemma 4.3.1(i) and

the cycle index Z (S n[S 2]), we get that

nO(Bn) =1

2 × 2nn!(

k|2n,k=2n

k2nk−1(

2n

k− 1)!|J k| + φ(2n)|J 2n|)

=

k|2n,k=2n

k2nk−1(

2n

k− 1)!

1

(2nk

)!

∂ 2nk (Z (S n[S 2]))

∂s2nk

k

|sk=0

+ φ(2n)∂ (Z (S n[S 2]))

∂s2n|s2n=0

Now we consider maps on the non-orientable surfaces with an underlying graph

Bn. Similar to the discussion of the Theorem 4.1, we get the following enumerationresult for the non-isomorphic maps on the non-orientable surfaces.

Theorem 4.3.2 The number nN (Bn) of non-isomorphic maps on the non-orientablesurfaces with an underlying graph Bn is

nN (Bn) =(2n − 1)!

n!+

k|2n,3≤k<2n

(2k)2nk−1(

2n

k− 1)!

∂ 2nk (Z (S n[S 2]))

∂s2nk

k

|sk=0

+1

2nn!(

s≥1n!

(n−

2s)!s!+ 4n(n

−1)!(

∂ n(Z (S n[S 2]))

∂sn

2 |s2=0

− ⌊

n

2⌋

)).




Proof Similar to the proof of the Theorem 4.3.1, applying the formula (1.3.3)in the Corollary 1.3.1 and the Lemma 4.3.1(ii), we get that

nL(Bn) =(2n − 1)!

n!+ φ(2n)

∂ n(Z (S n[S 2]))

∂sn2n

|s2n=0

+1

2nn!(s≥0

n!

(n − 2s)!s!+ 4n(n − 1)!(

∂ n(Z (S n[S 2]))

∂sn2

|s2=0 − ⌊n

2⌋))

+

k|2n,3≤k<2n

(2k)2nk−1(

2n

k− 1)!

∂ 2nk (Z (S n[S 2]))

∂s2nk

k

|sk=0.

Notice that nO(Bn) + nN (Bn) = nL(Bn). Applying the result in the Theorem4.3.1, we know that

nN (Bn) =(2n − 1)!

n!+

k|2n,3≤k<2n

(2k)2nk−1(

2n

k− 1)!

∂ 2nk (Z (S n[S 2]))

∂s2nk

k

|sk=0

+1

2nn!(s≥1

n!

(n − 2s)!s!+ 4n(n − 1)!(

∂ n(Z (S n[S 2]))

∂sn2

|s2=0 − ⌊n

2⌋)).


Fig. 4.2¸Calculation shows that

Z (S 1[S 2]) =s21 + s2

2

and




Z (S 2[S 2]) = s41 + 2s21s2 + 3s22 + 2s48

,

For n = 2, calculation shows that there are 1 map on the plane, 2 maps on theprojective plane, 1 map on the torus and 2 maps on the Klein bottle. All of thosemaps are non-isomorphic and same as the results gotten by the Theorem 4 .3.1 and4.3.2, which are shown in the Fig. 2.

§4. A classification of the closed s-manifolds

According to the Theorem 1.2.8, We can classify the closed s-manifolds by tri-angular maps with valency in

5, 6, 7

as follows:

Classical Type:

(1) ∆1 = 5 − regular triangular maps (elliptic);(2) ∆2 = 6 − regular triangular maps(euclid );(3) ∆3 = 7 − regular triangular maps(hyperbolic).

Smarandache Type:

(4) ∆4 = triangular maps with vertex valency 5 and 6 (euclid-elliptic);(5) ∆5 = triangular maps with vertex valency 5 and 7 (elliptic-hyperbolic);(6) ∆6 = triangular maps with vertex valency 6 and 7 (euclid-hyperbolic);(7) ∆7 =

triangular maps with vertex valency 5, 6 and 7

(mixed ).

We prove each type is not empty in this section.

Theorem 4.4.1 For classical types ∆1 ∼ ∆3, we have that (1) ∆1 = O20, P 10;(2) ∆2 = T i, K j, 1 ≤ i, j ≤ +∞;(3) ∆3 = H i, 1 ≤ i ≤ +∞,

where, O20, P 10 are shown in the Fig.4.3, T 3, K 3 are shown in the Fig. 4.4 and H iis the Hurwitz maps, i.e., triangular maps of valency 7.

Fig. 4.3¸




Fig. 4.4¸

Proof If M is a k-regular triangulation, we get that 2ε(M ) = 3φ(M ) = kν (M ).Whence, we have

ε(M ) =3φ(M )

2and ν (M ) =

3ε(M )

k.

By the Euler-Poincare formula, we know that

χ(M ) = ν (M )

−ε(M ) + φ(M ) = (

3

k −1

2

)φ(M ).

If M is elliptic, then k = 5. Whence, χ(M ) = φ(M )10 > 0. Therefore, if M

is orientable, then χ(M ) = 2, Whence, φ(M ) = 20, ν (M ) = 12 and ε(M ) = 30,which is the map O20. If if M is non-orientable, then χ(M ) = 1, Whence, φ(M ) =10, ν (M ) = 6 and ε(M ) = 15, which is the map P 10.

If M is euclid, then k = 6. Whence, χ(M ) = 0, i.e., M is a 6-regular triangulationT i or K j for some integer i or j on the torus or Klein bottle, which is infinite.

If M is hyperbolic, then k = 7. Whence, χ(M ) < 0. M is a 7-regular trian-gulation, i.e., the Hurwitz map. According to the results in [65], there are infiniteHurwitz maps on surfaces. This completes the proof.

For the Smarandache Types, the situation is complex. But we can also obtainthe enumeration results for each of the types ∆4 ∼ ∆7. First, we prove a conditionfor the numbers of vertex valency 5 with 7.

Lemma 4.4.1 Let Let C(T, n) be an s-manifold. Then

v7 ≥ v5 + 2

if χ(C(T, n)) ≤ −1 and

v7 ≤ v5 − 2

if χ(C

(T, n))≥

1. where vi

denotes the number of vertices of valency i in C

(T, n).




Proof Notice that we have know

ε(C(T, n)) =−χ(C(T, n))

13− 2

k

,

where k is the average valency of vertices in C(T, n). Since

k =5v5 + 6v6 + 7v7

v5 + v6 + v7

and ε(C(T, n)) ≥ 3. Therefore, we get that(i) If χ(C(T, n)) ≤ −1, then

13

− 2v5 + 2v6 + 2v75v5 + 6v6 + 7v7

> 0,

i.e., v7 ≥ v5 + 1. Now if v7 = v5 + 1, then

5v5 + 6v6 + 7v7 = 12v5 + 6v6 + 7 ≡ 1(mod2).

Contradicts to the fact that

v∈V (Γ) ρΓ(v) = 2ε(Γ) ≡ 0(mod2) for a graph Γ.Whence we get that

v7 ≥ v5 + 2.

(ii) If χ(C

(T, n))≥

1, then

1

3− 2v5 + 2v6 + 2v7

5v5 + 6v6 + 7v7< 0,

i.e., v7 ≤ v5 − 1. Now if v7 = v5 − 1, then

5v5 + 6v6 + 7v7 = 12v5 + 6v6 − 7 ≡ 1(mod2).

Also contradicts to the fact that

v∈V (Γ) ρΓ(v) = 2ε(Γ) ≡ 0(mod2) for a graph Γ.Whence, we have that

v7≤

v5−

2.

Corollary 4.4.1 There is no an s-manifold C(T, n) such that

|v7 − v5| ≤ 1,

where vi denotes the number of vertices of valency i in C(T, n).Define an operator Ξ : M → M ∗ on a triangulation M of a surface as follows:Choose each midpoint on each edge in M and connect the midpoint in each

triangle as shown in the Fig.4.5. Then the resultant M ∗ is a triangulation of thesame surface and the valency of each new vertex is 6.




¸Then we get the following result.

Theorem 4.4.2 For the Smarandache Types ∆4 ∼ ∆7, we have(i) |∆5| ≥ 2;(ii) Each of |∆4|, |∆6| and |∆7| is infinite.

Proof For M ∈ ∆4, let k be the average valency of vertices in M , since

k =

5v5 + 6v6v5 + v6 < 6 and ε(M ) = −

χ(M )13 − 2

k ,

we have that χ(M ) ≥ 1. Calculation shows that v5 = 6 if χ(M ) = 1 and v5 = 12 if χ(M ) = 2. We can construct a triangulation with vertex valency 5, 6 on the planeand the projective plane in the Fig. 4.6.

¸Fig. 4.6¸

Now let M be a map in the Fig. 4.6. Then M Θ is also a triangulation of thesame surface with vertex valency 5, 6 and M Θ

= M . Whence,

|∆

4|is infinite.




For M ∈

∆5

, by the Lemma 4.4.1, we know that v7 ≤

v5 −

2 if χ(M )≥

1 andv7 ≥ v5 + 2 if χ(M ) ≤ −1. We construct a triangulation on the plane and on theprojective plane in the Fig.4.6.

¸Fig. 4.7¸

For M ∈ ∆6, we know that k = 6v6+7v7v6+v7

> 6. Whence, χ(M ) ≤ −1. Since3φ(M ) = 6v6 + 7v7 = 2ε(M ), we get that

v6 + v7 − 6v6 + 7v72

+6v6 + 7v7

3= χ(M ).

Therefore, we have v7 = −χ(M ). Since there are infinite Hurwitz maps M onsurfaces. Then the resultant triangular map M ∗ is a triangulation with vertexvalency 6, 7 and M ∗ = M . Whence, |∆6| is infinite.

For M ∈ ∆7, we construct a triangulation with vertex valency 5, 6, 7 in the Fig.4.8.

Let M be one of the maps in the Fig.4.8. Then the action of Θ on M resultsinfinite triangulations of valency 5, 6 or 7. This completes the proof.

Conjecture 4.4.1 The number |∆5| is infinite.




Fig. 4.8¸



Chapter5

Open Problems for Combinatorial Maps

As a well kind of decomposition of a surface, maps are more concerned by math-ematician in the last century, especially in the enumeration of maps ([33] − [35])and graphs embedding on a surface ([22], [35], [53], [70]). This has its own right,alsoits contribution to other branch of mathematics and sciences. Among those mapapplication to other branch mathematics, one typical example is its contribution

to the topology for the classification of compact surfaces by one face, or its dual,one vertex maps, i.e., a bouquet Bn on surfaces. Another is its contribution to thegroup theory for finding the Higman-Sims group (a sporadic simple group ([6])) andan one sentence proof of the Nielsen-Schreier theorem, i.e., every subgroup of a freegroup is also free ([63] − [64]). All those applications are to the algebra, a branch of mathematics without measures. From this view, the topics discussed in the graphtheory can be seen just the algebraic questions. But our world is full of measures.For applying combinatorics to other branch of mathematics, a good idea is pullbackmeasures on combinatorial objects again, ignored by the classical combinatorics andreconstructed or make combinatorial generalization for the classical mathematics,such as, the algebra, differential geometry, Riemann geometry, Smarandache ge-

ometries, · · · and the mechanics, theoretical physics, · · ·. For doing this, the mostpossible way is, perhaps by the combinatorial maps. Here is a collection of openproblems concerned maps with the Riemann geometry and Smarandache geometries.

5.1 The uniformization theorem for simple connected Riemann surfaces

The uniformization theorem for simple connected Riemann surfaces is one of those beautiful results in the Riemann surface theory, which is stated as follows([18]).

If S is a simple connected Riemann surface, then S is conformally equivalent to

one and only one of the following three:(a) C ∞;(b) C;(c) = z ∈ C||z| < 1.

We have proved in the Chapter 2 that any automorphism of a map is conformal.Therefore, we can also introduced the conformal mapping between maps. Then, how can we define the conformal equivalence for maps enabling us to get the uniformiza-tion theorem of maps? What is the correspondence class maps with the three type(a) − (c) Riemann surfaces?



Chapter 5 Open Problems for Combinatorial Maps 89

5.2 The Riemann-Roch theorem

Let S be a Riemann surface. A divisor on S is a formal symbol

U =k

i=1

P α(P i)

i

with P i ∈ S , α(P i) ∈ Z. Denote by Div(S ) the free commutative group on thepoints in S and define

deg U =k

i=1

α(P i).

Denote by H(S ) the field of meromorphic function on S . Then for ∀f ∈ H(S ) \0, f determines a divisor (f ) ∈ Div(S ) by

(f ) =

P ∈S

P ordP f ,

where, if we write f (z) = zng(z) with g holomorphic and non-zero at z = P , thenthe ordP f = n. For U 1 =

P ∈S

P α1(P ), U 2 =

P ∈S P α2(P ), ∈ Div(S ), call U 1 ≥ U 2 if

α1(P ) ≥ α2(P ). Now we define a vector space

L( U ) = f ∈ H(S )|(f ) ≥ U , U ∈ Div(S )Ω( U ) = ω|ω is an abelian differential with (ω) ≥ U.

The Riemann-Roch theorem says that([71])

dim(L( U −1)) = deg U − g(S ) + 1 + dimΩ(S ).

Comparing with the divisors and their vector space, there ia also cycle space andcocycle space in graphical space theory ([35]). What is their relation? Whether can we rebuilt the Riemann-Roch theorem by map theory?

5.3 Combinatorial construction of an algebraic curve of genus

A complex plane algebraic curve Cl is a homogeneous equation f (x,y,z) = 0in P 2C = (C 2 \ (0, 0, 0))/ ∼, where f (x,y,z) is a polynomial in x, y and z withcoefficients in C. The degree of f (x,y,z) is said the degree of the curve Cl. Fora Riemann surface S , a well-known result is ([71])there is a holomorphic mapping ϕ : S → P 2C such that ϕ(S ) is a complex plane algebraic curve and

g(S ) =(d(ϕ(S )) − 1)(d(ϕ(S )) − 2)

2.




By map theory, we know a combinatorial map also is on a surface with genus.Then whether we can get an algebraic curve by all edges in a map or by makeoperations on the vertices or edges of the map to get plane algebraic curve with given k-multiple points? and how do we find the equation f (x,y,z) = 0?

5.4 Classification of s-manifolds by maps

We have classified the closed s-manifolds by maps in the Section 4 of Chapter 4.For the general s-manifolds, their correspondence combinatorial model is the mapson surfaces with boundary, founded by Bryant and Singerman in 1985 ([8]). The

later is also related to the modular groups of spaces and need to investigate furtheritself. The questions are

(i) how can we combinatorially classify the general s-manifolds by maps with boundary?

(ii) how can we find the automorphism group of an s-manifold? (iii) how can we know the numbers of non-isomorphic s-manifolds, with or with-

out root? (iv) find rulers for drawing an s-manifold on a surface, such as, the torus, the

projective plane or Klein bottle, not the plane.

The Smarandache manifolds only using the triangulations of surfaces with vertex

valency in 5, 6, 7. Then what are the geometrical mean of the other maps, such as, the 4-regular maps on surfaces. It is already known that the later is related tothe Gauss cross problem of curves([35]). May be we can get a geometry even moregeneral than that of the Smarandache geometries.

5.5 Gauss mapping among surfaces

In the classical differential geometry, a Gauss mapping among surfaces is definedas follows([42]):

Let S ⊂ R3

be a surface with an orientation N. The mapping N : S → R3

takesits value in the unit sphere

S 2 = (x,y,z) ∈ R3|x2 + y2 + z2 = 1along the orientation N. The map N : S → S 2, thus defined, is called the Gaussmapping.

we know that for a point P ∈ S such that the Gaussian curvature K (P ) = 0and V a connected neighborhood of P with K does not change sign,

K (P ) = limA→0

N (A)

A,




where A is the area of a region B⊂

V and N (A) is the area of the image of B bythe Gauss mapping N : S → S 2. The questions are

(i) what is its combinatorial meaning of the Gauss mapping? How to realizes it by maps?

(ii) how can we define various curvatures for maps and rebuilt the results in theclassical differential geometry?

5.6 The Gauss-Bonnet theorem

Let S be a compact orientable surface. Then

S

Kdσ = 2πχ(S ),

where K is Gaussian curvature on S .This is the famous Gauss-Bonnet theorem for compact surface ([14], [71] − [72]).

This theorem should has a combinatorial form. The questions are(i) how can we define the area of a map? (Notice that we give a definition of

non-Euclid area of maps in Chapter 2.)( ii) can we rebuilt the Gauss-Bonnet theorem by maps?

5.7 Riemann manifolds

A Riemann surface is just a Riemann 2-manifold. A Riemann n-manifold (M, g)is a n-manifold M with a Riemann metric g. Many important results in Riemannsurfaces are generalized to Riemann manifolds with higher dimension ([14], [71] −[72]). For example, let M be a complete, simple-connected Riemann n-manifoldwith constant sectional curvature c, then we know that M is isometric to one of themodel spaces Rn, S Rn or H Rn. There is also a combinatorial map theory for higherdimension manifolds (see [67] − [68]). Whether can we systematically rebuilt theRiemann manifold theory by combinatorial maps? or can we make a combinatorial generalization of results in the Riemann geometry, for example, the Chern-Gauss-

Bonnet theorem ( [14], [37], [71])? If we can, a new system for the Einstein’s relativetheory will be found.



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automorphism groups of maps, surfaces and smarandache geometries, by linfan mao

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